US1263008A

US1263008A - Bevel-gearing.

Info

Publication number: US1263008A
Application number: US2302815A
Authority: US
Inventors: Harvey D Williams
Original assignee: GEAR IMPROVEMENT Co Inc
Current assignee: GEAR IMPROVEMENT COMPANY Inc; GEAR IMPROVEMENT Co Inc
Priority date: 1915-04-22
Filing date: 1915-04-22
Publication date: 1918-04-16
Anticipated expiration: 1935-04-16

Description

H. D. WILLIAMS.

BEVEL GEARiNG. APPLICATION FILED APR. 22. 1915.

Patented Apr. 16, 1918.

5 SHEETS-SHEET I 11100726021 Harneyfl William; 7 z' 1/ H. D. WILLIAMS.

BEVEL GEARING.

APPLICATION FILED APR. 22. 1915.

Patented Apr. 16, 1918.

5 SHEETS-SHEET 2.

7 W Harveyfl William};

H. D. WILLIAMS.

BEVEL GEAR ING. APPLICATION FILED APR. 22. 1915.

Patented Apr. 16 1918.

5 SHEETS-SHEET 4.

Winedgefi H.D. WlLLlAMS.

BEVEL GEARING.

APPLICATION FILED APR. 22. 1915.

m N na I n e. N h Am W MW fl. mm W H 6 a m h u B om M 0 e M 4w 3 e 6 a UNILED PATENT @Fir HARVEY D. WILLIAMS, 0F WALLINGFORD, CONNECTICUT, ASSIGNOR TO GEAR IMPROVE- MENT COMPANY, INC, OF NEW YORK, N. Y., A CGRPORATION OF NEW YORK.

BEVEL- GEAEING.

To all whom, it may concern Be it known that I, IIARVEY I). W ILLrAMs, a citizen of the United States, residing at VVallingford, in the county of New Haven and State of Connecticut, have invented certain new and useful Improvements in Bevel- Gearing, of which the following is a specification.

My present invention relates more especially to the master-form bevel-gearing described in Letters Patent of the United States No. 1,112,509, granted to me October 6, 1914. A principal object of the present invention is to furnish bevel gears of the same master-form class (of which one instance is set forth in my said patent, another instance being herein illustrated), and in which the respective pairs of gear-tooth working-faces shall have improved structural features and operational relations, as hereinafter more fully explained.

For convenience, and to facilitate a description of my present improvements, 1 have herein employed the term wheel for designating the bevel-gear having the teeth thereof Provided with working-faces of a configuration adapted to be made by and according to the singlereproduction method, and have designated the mating gear as the pinion, without regard, however, to their relative actual sizes but having in mind that usually the wheel is larger than the pinion.

WVhile the present improvements are not intended for use in bevel-gears of some pro portions and sizes, these restrictions in the range of applicability are compensated for and offset by important advantages, in mode of operation and facility of manufacture, and thus I utilize to the advantage of a large class of cases,especially in precisionized gearing,certain features not equally applicable to the whole range of bevel-gearing.

One object of my present improvement is to furnish bevel-gearing of the aforesaid master-form class, in which the width of the addendum portions of the working faces of the pinion teeth may be considerably increased at the outer ends, relatively to the width of said surfaces as measured at the inner end of the teeth, and withoutincreasing the otherwise normal height of the wheel-teeth at the inner ends thereof; also,

Specification of Letters Patent. Patented A1313 16 1918. Application filed April 22, 1915.

Serial No. 23,028.

in this kind of bevel-gearing, to increase the relative width, at the outer ends of the wheel-teeth, of that portion of their working surfaces which will operate as dedendum surfaces or flanks.

In the accompanying drawings forming a part of this specification, I have illustrated the principles and structural features of the gearing in connection with geometrical representations of the wheel-including sphere and certain of its great-circles, thus follow ing a well-known graphical method.

Figure 1 is a side view,shown partially in section,0f a pair of gears comprising the bevel-gear wheel B, and pinion P, together with geometrical delineations of the wheelincluding sphere, D, and of certain circles and lines (including the great-circles 2 and 4:), relating to the derivation of the master-form for the tooth-faces of the wheel, and for the location of one pair of these faces.

Fig. 2 is an oblique side view of the principal features illustrated in Fig. 1, but with the pinion P indicated by dotted lines, as seen from the direction of the arrow 7, in Fig. l.

Fig. 2 is a fragmentary view, on an enlarged scale, for further illustrating the features shown in the central portion of Fi 2, particularly as to the location on the sphere D of the profiles of the wheel and pinion teeth; this view, to facilitate comparison, is drawn with the master-form aXis 50 in alinement (horizontally) with the same axis which in Fig. 2, is shown on the pitch-line 7 Fig. 2 Fig. 2, and Fig. 2 are a series of diagram views arranged for illustrating the master-form feature of this system of gearing, and also the counterpart feature of such master-forms as applicable to the wheel, B, by the single-reproduction, and to the pinion, P, by the compound-reproduction. as hereinafter more fully explained.

Fig. 3 is an oblique and diagrammatic view for more clearly exhibiting the pinion construction, this being more fully shown in subsequent views supplemental thereto.

Fig. 4 is an oblique and diagramn'latic view,on an enlarged scaie,corresponding with Fig. 3,-of a portion of the wheel B when made in accordance with Figs. 2, 10 and 14, and the other views and diagrams supplemental thereto, as hereinafter explained.

tain features hereinafter explained; this view comprises diagrammatic lines for comparison with other views, especially with Figs. 3, 4 and 10, and the diagrams supplemental thereto.

Fig. 5 is a diagrammatic and fragmentary view drawn in alinement with, and supplemental to Fig. 5, for showing certain details as seen from the left-hand in F 5.

Fig. 6 is a view similar in character to Fig. 5, and is drawn below and in alinement with said Fig. 5, to facilitate comparison; and for more fully illustrating certain features and relations of the masterfo-rm construction of the wheel member of the pair of gears, including a complementform, and two arrangements of longitudinal parallelism as hereinafter explained.

6 is a view similar in character to Fig. 5 and is drawn in alinernent with both Fig. 6 and Fig. 5 for illustrating certain wheel-tooth details as seen from the lefthand in Fig. 6, and also for showing these in comparison with corresponding piniontooth details shown in Fig. 5*.

Fig. 7 is a diagram, drawn on an enlarged scale, for more fully bringing together and illustrating in a single view, certain features and relations which obtain at both the outer and the inner ends of the wheel tooth-faces, and which are indicated, but in a less complete manner, in Figs. 2, i. 6 and 6*, as hereinafter more fully explained.

Fig. 8 is a supplemental diagram comprising a plan view of a pair of wheel-teeth as seen from above in Fig. 6, one of the teeth being shown in section as if the addendum portion were removed down to the surface of the pitch-cone, this cone being indicated by the line ac in said Fig. 6, and the outline of the sectional area being a symmetrical and quadrilateral figure; this view, F ig. 5, also illustrates how the toothfaces may be considered as arranged in two series of skew-located faces.

Fig. 9 is a view similar to the left-hand portion of Fig. 6, but drawn on a larger scale, for comparison with other and adjacent figures of the drawings.

Fig. 9 is a diagram or fragmentary view showing a wheel-tooth in a position as shown in Fig. 2. and also as seen from the lefthand in Fig. 9; this view, Fig. 9 further illustrates certain operational features of the meshing wheel and pinion, and certain repinion-tooth, as 9, into working position.

relatively to the wheel-teeth.

Fig. 10 is a view supplemental to and drawn in alinement below Fig. 9, for showing a pinion-tooth in the symmetrical position (as in Fig. 5 and in a meshing posi tion between a pair of adjacent wheel-teeth, as seen from the left-hand in Fig. 10.

Figs. 11 and 12 are diagrammatic views illustrative of the preferred method of making the wheel member for a pair of the gears when these are constructed with the body of the wheel -tooth located between outwardly-converging tooth-surfaces arranged in. the longitudinal-parallelisms as hereinafter set forth.

Fig. 13 is a diagrammatic view illustrative of a preferred method for generating the work-surfaces of tie pinion teeth, as hereinafter explained.

Fig. 14: is a side view, chiefly diagrammatic, of the pinion P, as seen from the lefthand in Fig. 13; these two views being drawn in alinement to facilitate comparison.

Fig. 15 is a face view of a. portion of the pinion as seen from above in Fig. 1-1, with this diiference however, that the tooth-zone is here shown as if the pitch-surface, were unrolled into a plane, for thereby bringing all the pinion-teeth to stand as if vertically disposed in a plan view, to facilitate descrintion and comparison.

Fig. 16 is an enlarged fragmentary view showing pinion teeth corresponding to those indicated in Fig. 15. as seen in the direction of the arrow 1* in Fig. 14- and showing these teeth as arranged with the longitudinallyparallel construction applied to and between the adjacent working-faces of adjacent pinion-teeth, thereby giving to these teeth a longitudinally-tapering formation, as measured upon the surface of the geometric.

pitch-solid, here indicated as a cone.

Similar characters designz-ite like parts in all the views.

In this improved bevel-gearing, a pair of bevel-gears, as B, P. (see Fig. 1) will usually comprise a conical wheel having plane-surface tooth-faces, as f 7, arranged in parallel with an axis, a], which at its outer end coincides with the intersection of two great-circles. as :2 and 4, of a wheel-including sphere, D. relatively to the center, 0, of which said axis may have, in some instances, a skew-location. l he said tooth-face planes also lie in the planes of two circles, as

'3 and 5, which, respectively, may be ofl'set outwardly from said great circles in such a manner (Fig. 2) as to make the longitudinal, or lengthwise parallelism apply to the pairs of faces between which the wheelteeth are located; in these tooth-face pairs, the two face surfaces are also outwardlyconverging. For convenience of comparison, Figs. 1 and 2 have diagrammatic lines, and reference characters therefor, similar to corresponding views in my aforesaid Letters Patent.

In the preferred form and arrangement herein selected for illustration, the inwardly-converging pair of tooth-surfaces are coincident with the geometric planes f f Figs. 2 6 and, have, in the wheel B, a series of parallel geometric surface-elements which are also in parallel with a master-form-axis, as, that is located on an angle, 5, Figs. 5 and 6 relatively to a pitchcone surface-element, which is radial to the center a, this angle, for convenience, l designate as the depression angle. Said tooth-surface planes also have another series of parallel geometric surface-elements which are in parallel with the line of intersection, of said inwardly-converging geometric planes, and with a complemental masterform axis, as y, having an elevation-angle, b and located (as seen in side view, Fig. 6), for intersecting said depression-angle axis-line, w, as and for the purposes hereinafter more fully explained.

In a pair of my improved bevel gears, the wheel member, 13, is regarded as a master wheel, for the reason, among others, that a series of pinions of various sizes and pitch-cone angles, respectively,and with similar or different axial positions,inay be made to properly mesh with and drive, or be driven by, a single wheel. The profileangle of the wheel-teeth corresponds with a geon'ietric figure (as F, Fig. 2) which I designate as the master-form and which in passing along a given line, generates a geon'ietric solid that comprises a pair of transversely converging surfaces, preferably in the form of true planes, and having surface-elements parallel to each other and to a certain intermediate line which is a master-form-axis. A space-forming tool, as J, Figs. 11, 12, 18, (as one of a pair of counterpart tools,see Fig. 2 may have the edges thereof located in coincidence with the side lines, of such a masterform, so that the tool when passing along said master-form-axis, will in effect reproduce and apply the master-form, F, througnout the length of the tooth, and thus give to the tooth-face surface-elements a peculiar longitudinal pa allelism. lhese terms and others as used herein, are more fully explained in my aforesaid Patent No. 1,112,509, together with descriptions of the transverse, the inward and the outward, convergence,-also, of the various skew features and relations, and the meshingaction of the several tooth-faces,includthe progressive feature thereof,as involved or resulting from such parallelism, or from the single and compound methods of reproductions. In these opera tions, as herein illustrated, the surfaces of the teeth of wheel 13 are developed by the tool 1 used as a single-reproduction tool, and the surfaces of the teeth of pinion P are generated by the tool J used as a generating tool.

In the master wheel, the plane-surface tooth-faces are not only skew-located as explained, but are also non-radial, since in any case no more than one surface-element line of any said tooth-face,and not the surface, as such,can lie in a plane of the wheel-axis; and this arrangement can only exist when the skew-angle of the master fornraxis corresponds with the tooth-space angle, which, in practice may seldom ccur.

In my present improvements, and as also described in my said prior patent, the method single-reproduction is or may be applied to the making at one time of a pair of adjacent tooth-faces. Also in this operation, the tool. preferably employed therefor, comprises two straight-line cutting edges, each of which moves in the geometric surface in which a wheel-tooth face is located. In the present instance, also, the said pair of cutting edges, as al u Figs. l1, 12, are arranged for operating simultaneously on a forward tooth-face and a rearward toothface, but in the present instance I have shown said cutting edges arranged for so making these tooth-faces on two adjacent wheel-teeth, respectively. For instance, in Fig. 11, the tool-edge 0 operates on the right-hand side 7 of one tooth, while the opposite tool-edge v operates on the lefthand side, 7, of an adjacent tooth. Thus the application of one form of counterpart tool is herein shown arranged for making a pair of wheel-tooth workin -faces, in the same manner as a similar counterpart tool, J, is employed for making the corresponding pair of working-faces on the ad]acent sides of two adjacent pinion teeth,see Fig. 13 in the accompanying drawing, and Fig. 25 in my said prior patent. The direc tion of movement, however, of the space forming tool is materially different when employed in the making of wheel-tooth working faces which are shaped and arranged as herein illustrated.

The present method of forming the workingsurfaces of the gear teeth, is in the nature of an improvement on the mode of making the tooth-surfaces of the wheel and pinion which is described in my U. S. Patent- No. 1,112,509; and, besides securing similar results, as therein described, the present improvement also provides for making the pairs of wheel-tooth faces with equal economy while making these faces of increased depth at the outer end thereof and below (within). the pitch-cone. This additional tooth-face area,as f 4,-may be regarded as beingin the nature of a triangular extension of a normal rectangular tootl1face area the lower edge of which is in alinement with the master-form-axis. Such a normal tootlrface area is in icated by the dimension in Fig. 1.

By making the wheel, B, with each of the teeth, as 71/, b (Fig. 4), thereof located between an outwardly-converging pair of the geometric master-form surfaces (herein' shown as planes), it becomes possible in making the conjugate-curvature of the pinion tooth-faces, to employ a double-profile tool, as J, Figs. 2 and 13, for operating on two pinion tooth working surfaces at the same time, so that, in practice, the usual duplicate operations of finishing the crosssectional curvature of the pinion-teeth on one face at a time, (as heretofore practised when these teeth are longitudinally tapering), are in this system combined togetl1er,--two such single operations into one duplex operation,which is a compoundreproduction and results, as will be evident, in doubling the otherwise normal capacity of the pinion-tooth-forming machine, and in a proportionate reduction in the cost of manufacture.

lVhen the angular position of the masterfor 1 profiles for the wheel and pinion are found, thus locating the position in the wheel of the respective tooth-surface planes, an in vardly-converging pair of these planes on being extended will intersect on a line, as 3 Figs. 6 and 6 which will be the line of movement of a tool (as T, Fig. 12), the cutting edges of which coincide with profile lines of said inwardly-converging pair of tooth-surface planes, when the cutting edges are also located in a plane at right-angles to said line of -movement. This feature is shown by Figsr2 and 6*, where the inwardly-converging planes 7, J, are extended to an intersection on said line if, which thus becomes the line of movement for a tool having the cutting edges thereof, coinciding with said planes, respectively.

The geometric surfaces with which the outwardly converging tooth faces of the wheel coincide, will be seen as straight lines, as at 7, f Figs. 2 6 when the masterform profiles are straight lines and are symmetrically arranged; these surfaces will then coincide with the planes of two intersecting small circles, (as 3 and 5 Figs. 1 and 2) which, in the surface of said wheel-including sphere, are parallel to two intersecting great-circles, as 2, 4., and therefore the line of intersection, as 1 Figs. 2, 2 7, of the planes of said small-circles will be a straight. line.

For convenience of illustration, and also conforming, to a well-known conventional practice, (as employed, for instance, in the well-known Tredgolds approximation method of delineating bevel gears) I have herein generally shown the profile lines, at the ends of the tooth-surfaces, (excepting in Figs. 1 and 2), as if they were located in the surface of a geometric cone which is tangential to the wheel-including sphere at the pitch-circle thereof. As regards the wheel B, the position of one side of such a 1 cone is indicated by the dotted line 0 in Fig. 1. The line 0 may also be considered as indicating, in projection, a plane surface in which the view-Fig. 2 is supposed to be drawn, a line of sight vertical to such surface being indicated by the arrow 7 The two master-form axes w, m, of the wheel B, and pinion P, respectively, may be considered as coordinate in character, since each of saidaxes, respectively, has a similar relation to a pair of tooth-faces, and since said axes also come into coincidence when those pairs of tooth-faces come into the symmetrical position; such a position is shown in one arrangement in Fig. 9, where the master-form-axis of a pair of the piniontooth faces, coincides with the similar axis as of a coacting pair of the wheel-tooth faces. in Fig. 10 said axes :0, 00 are shown one pair approaching and one pair receding from the instant axis position, which is here shown at the point y g". r

The location, or position, of the line m, Fig. 6, which indicates the plane in which the wheel-tooth is of uniform thickness as shown in Fig. 8, will, of course, be determined by, and should correspond with, the position in the mating pinion P, of the corresponding lines, as w, representing the direction of movement of the tool (as J, Fig. 13), when the configuration of the workingfaces of the pinion, (see Figs. 13 to 16) is generated by a cutting process, and according to the compound-repreduction configuration.

From the foregoing explantions, it will be understood that in my present improvements, the wheel member of he pair of gears, has two kinds, or arrangements, of parallelism, one of these arrangements is a parallelism relatively to a geometric masterform-axis as 00, (see Figs. 6, 6 and 9, 9 which appertains to the wheel-tooth when this is located between the outwardly-converging pair of geometric planes, and which has a parallel relation also to a corresponding master-form-axis, as m, of the toothspace of a meshing pinion. Another kind or arrangement of parallelism relates t 'the pairs of tooth-surfaceplanes which are in- 1,2ss,oos

wardly-converging, and between which the tooth-spaces are located. (See Figs. (3 to This arrangement or parallelism applies more especially to the gear wheels when these have tooth-spaces which as measured on the pitch-cone, are of a tapering width, and are wider at the outer end than at the inner end; this is shown, for instance, in Figs. 4, 6, 9 10. In practice, the features here explained, result in an important advantage, since the tool-marks usually left on the tooth-faces of the wheel and pinion, respectively, in the finishing operations, are thereby located in such relatively oblique directions as to minimize the frictional adhesion otherwise normally occurring when the tooth-surfaces are newly made; this tends to materially improve the mode of action, and the wearing down of the freshly machined surfaces to a perfect running condition.

By making said taper of space-width (when measured as here indicated) equal to the angle guotended by the pitch-arc (as indicated by the angle 7 Fig. 8) the described parallelism of the tooth-surface elements would bring them arallel, also, to the master-form-axis when this is located in the surface of the geometric pitch-cone. But I have herein illustrated that parallelism with relation to the master-form-axis, when thi is located not in said surface of the pitclicone, but is located well within and therefore deviates from such pitch-con geometric surface. This deviation, in practice, may vary in amount in diflerent instances. and may also be variable in direction. In Figs. 5 and 6, for instance, said deviation is only in a plane radial to the wheel axis, and is measured by the deviation of the lines 7 and a, while the outer end of said master-form-axis a; coincides at 7 (Fig. 6) with a pitch-cone surface-element, as 7, which is also indicated in said radial plane. In other instances, said master-formaxi ma have a skew relation (not shown herein) to the wheel-axis,and so not intersect said wheel axis,-for thereby making the wheel with the teeth and tooth-spaces thereof angularly-disposed on the tooth-zone of the wheel, as more fully set forth in my aforesaid Patent No. 1,112,509.. And in such an angularly-disposed form or arrangement, the master-form-axis, in some instances, may be somewhat curved,especially(and preferably) on a circular curvature,but this feature is not specifically claimed herein, and constitutes in part the subject-matter of my copending application Serial No. 853,017.

In the larger diagrams Figs. 9 and 10, illustrating successive positions of the comeshing pinion teeth 9, and wheel-teeth h, 1 have designated successive wheel teeth as 1 2 72, h and the intervening wheel-toothgpaces 7L5 are successively indicated by hs ha Similarly, by thus employing combined characters, the pinion-toothspaces, gs, are SllCCQl-i iyely indicated by (/8 8 983. The master-rorin-axis, w, of the wheel-teeth is distinguished, in the successive wheel-teeth, respectively, by combining the reference characters, as aih [0723, 5072. and, similarly the corresponding master-form axes, Q1, of the pinion tooth-spaces 98, are successively indicated by xgs (Mg s mgs respectively. ll hen a wheel-tooth, [2 is in the symmetrical. position (Fig. 9), the master-form axis a, here designated by self, coincides with the correspondingmaster-form axi :0, here designated by arr s And, similarly, when a pinion-tooth, as comes to the symmetrical position (Fig. 10), the complement-form axis, y, (of a wheel tooth-space, as ho here designated by hs comes into coincidence with corresponding piniontooth axis, 7 here designated by yg Thus a pair of the master-form axes a, so come into coincidence in one )OSltlOll of the wheel and pinion, (Fig. 9 and a pair of the complementforni axes, 3 y," come into a similar coincidence in another posititon of tl e wheel and pinion (Fig. 10) but these coincidences occur at the same position, or plane, of sym metry.

As will now be evident on comparison of Figs. 5, to 10, the series of wheel-tooth faces may be said to be arranged in successive and alternating pairs and in two series of pairs, in which the faces of each of the pairs of one series are outwardly-converging while in the pairs of the other series th said faces are inwardly-converging, each said tooth-face being included in a pair of each of said series. Each outwardly-com verging pair of the wheel-tooth faces are arranged in parallelism with relation to a masterform axis, 02, which may be said to substantially coincide with or intersect the pitch-surface at or near the outer ends of the wheel teeth, (see line 7, Figs. 1, 6 9, 10") while each inwardly-converging pair of said faces are arranged in parallelism with rela tion to a complement-form axis, y, so located as to coincide with or inter-et the pitclrsurface (indicated by the line 7. Fig. 6), at or near the point 6 (Fig. 6) at the inner ends of the wheelteeth, and so located that the cone-of-revolution of the complement-form axis, 3 (Fig. 6), has or subtends an apex-angle, as 2 which smaller than the apexangle 2 of the cone-of-rcvolution of the master-form axis, :0. In Fig. 6, the pitch-surface of the wheel B is indicated by the cone (or geometric solid) of revolution,

2 and in Fig. 5, the corresponding pitchsurface of the pinion P is indicated by the cone (or geometric solid) of revolution, .2 when the wheel and pinion are brought into full mesh. as indicated in Figs. 1. 9 and 10, these solids-of-revolution, (which in the usual tin-skewed forms of bevel gears, will be, of course, true cones of revolution), will come into the relation of two contacting revolving geometric bodies arranged as it each one rolls upon the other. I

In the manufacture of precisionized bevelgearing, it is a matter of practical importance that, in the pinion, the several conesot-revolution, including a dedendum cone 2, the geometric pitch-cone 2 and the addendum-cone .2 shall have coincident apexes, as at '0, Fig, 5. This feature of apex coincidence is deemed to be especially important and useful in the making of pinions for my improved gearing, since in this system only the tooth-races or the pinion are produced by the method of compound-reproduction, while the wheel member ofthe pair is made by the more simple and much cheaper method of single-reproduction. Besides the important advantage of the coincident coneapexes, this construction also provides for materially enlarging the tooth-face workingsurt'ace by reason of the described increasing height of the addendum from the inner end toward the outer end, as shown in Figs. 5, 10 and 1%; this latter feature, however, involves the provision of a correspondingly increasing depth of wheel tooth-space, outwardly from the inner end thereof, as indicated, for instance, in Figs. 6 and 9, and aselsewhere herein more fully explained.

For accomplishing the foregoing objects, especially as regards the Pinion, P, I have indicated in Fig. 5, a construction in which he intersection of the axes m and :0 of Wheel 13 and pinion P, respectively, takes place at the point 0, and have selected this point as the place where several of the said cones-of revolution have their apexes in coincidence; in this view the wheel-teeth and pinion-teeth are understood to be shown as having a radial position, (commonly so called), and the gear-wheels as having their axes in one plane. Under these conditions, the pitch-cone 2, has .traightdinc surfaceelement extending from the center 0 to the peripheral point 3 (Fig. 5) and this line when in the symmetrical position, (Fig. 5), Will coincide with the instant-axis. The dedendum height, as 4.4 being new determined, the line 6 isdrawn from the point 6 to said center 0, then the line, 6 is one side of a cone, 2, which is the cone-otrevolution of said lower line 6 of a pinion tooth-space, and has its apex in said center point,-c. Similarly, the addendum height a being known or assumed, the line 6 drawn from the point e to said center 0, and thus delineates one side of a Gone .2 which is the addendum cone, and which also has the apex thereof at or in said common. center 0. Thus the three said cones-of-revolution e .2 and 2 may have coincident apexes all located at one given polnt, which meshes point, in practice, may be located at the intersection of the said wheel and pinion axes m and w in the construction here particularly described,it will be evident that the working surfaces of the teeth, of both wheel and pinion, are made up of two areas, and these areas may be said to be kinematically, or functionally distinct. One of said areas. in the wheel-tooth surface, (Fi 9), is the addendum-face t, lying between the addendum-cone line 6 and the line as of the master-form cone 2 (Fig. 6) and therefore 1ying outside of said cone 2"; normally, or usually, this face is substantially or nearly a parallelogram. The other said Wheel-tooth area is the tapering dedendum-face 7 lying between the said line so of the masteri orm cone and the line 6 of a complementtorm cone, 2 Fig. 6) which has a greater length (of side lines) and a smaller coneangle,as e ,than said master-form cone .2 or the pitch cone .2 Figs. 5 and 5, being (lawn in projection,that is, in alinement,theretore the dotted line represent ing the axis-line 3 is shown extending from the point or position y in Fig. 5 to the point 0 in Fig. 5; the other axial line an, however, terminates at the cone-apex represented at the point 0, in Fig. 5. In apply ing the master-form feature to the production of tooth-faces,as f f (Figs. 5 9),-in the preferred arrangement shown in Fig. 5, I locatethe masterdor'm axis y with the outer end in the pitch-circle of the pinion and parallel to the dedendum line This brings said axis line '3 (when the pinion and Wheel are brought together accordance with the diagrammatic illustration in Fig. 6), to the wheel-axis :0 at a point, 0 (Fig. 6) which is within'the pitch cone 2 of the coacting wheel, and outside of the pitch-cone .2 of the pinion, by the distance 10', which is equal to the ded'endum heightu Fig. 5. In generating the toothtaces of the pinion the line-ot-movement of the tool being along and parallel to said axis-line :0, while the cutting-edges of the tool are straight lines, therefore these edges follow in geometric planes and the pinion tooth-faces will be so generated and shaped or curved throughout their length as to conform to a configuration of tooth-surface in which the profiles at all points in the'length thereof will be conJugate to corresponding profile lines in the said geometric planes orsurfaces.

In Fig. 5 and considering the point 0 as a center (or point of intersection), the angular position of the pitch-cone line z/c, rela tively to the master-form axis 00 is indicated by said distance U and for convenience I designate the angle thus subtended the master-form depression-angle; this angle 0, y, 0 (Fig. 5) corresponds, as will be evident,

of the pair of profile lines h f and it to the position in the wheel, of the coordinate master-form-axis as m and also to the parallelism in the tooth-face planes of the wheel, of one series of surface-elements relatively to such depression angle and the master-form-axis m.

In Fig. 9 (as in Fig. 2), three successive wheel teeth are designated by h k 72?, respectively, and are shown located between duplicate arrangements of master-form planes of which one pair as seen in end view are indicated in Fig. 2. Each of these pairs of profiles corresponds, of course, with the master-form F, Fig. 2". Since these master-form outlines are positioned with relation to the wheel-including sphere,-as D, Figs. 1 and 2the several center lines, I), 5 6 respectively, correspond substantially in relative position with longitudelines on such sphere.

The master-form outline F, of Figs. 2 2 and 2, is shown applied in Figs. 6% 9 and 10", to the successve wheel-teeth 7L1, 7L and 71. which are shown arranged symmetric-ally with relation to the longitudelines 5 5 and 7), respectively, and in an inverted position relatively to the said Figs. 2 2 and 2. In Fig. 10 the sidelines, or tooth-end profiles and 7f of the wheeltooth 7L2 correspond in angular position or arran ement with the pair of side-lines of the master-form, this angle being indicated by 12. Similarly, the corresponding 0011'].- plement-fornris indicated in Fig. 10 by the angular positions (relatively to each other) this angle being indicated by 2 By comparing these forms, lines and angles, it will be evident that, in practice, the relative positions of lines I i and k will very nearly correspond with the relative positions of the longitude-lines 5 and Z2 which may be said to indicate meridian planes of the nreel-teeth 7L2 and it respectively; so that the angle of line b relatively to the line 5 (where these lines cross the pitch-line 7), added to the angle 2 should be substan tially equal to the angle 2 These angles in any given instance, will depend, it will be evident, on the location of the pitch-circle relatively to the poles and to the equatorial circle of the wheelincluding sphere. In practice the described tool-angle corresponding to 2 Fig. 10, may be obtained by a direct measurement of the tooth-surface. planes, or by computation. In 9 the wheel-tooth 71. represents (by its profile lines 72. 7 and 11 the master-form as located between the pair of pinion tooth-surfaces f and g f which are conjugate to said wheel-tooth surfa respectively, as further indicated by the same two pairs of prolile lines as shown in Fig. 13, which is elsewhere herein more fully explained.

The axial line, as y, of the complementform is indicated as being so located (as indicated by the position of this line in Fig. 10), relatively to the inaster-form-axis a, that the two axial lines 00 and g subtend an angle 2 (which I designate as the formaxis angle), which is greater than the ad dendum angle 2 (see Fig. 5) of a pinion having the pitch-cone and the addendum cone thereof with coincident apexes already described. And this excess of the form-axis angle 2 over the said addendum angle .2 is preferably made suiiiciently large for thereby bringing a pair of inwardly converging complement-form planes (geometric surfaces) into coincidence with the adjacent geometric planes (surfaces) of a pair of master-form geometric solids (Fig. 7) according to which. the pinion tooth working surfaces are generated.

The line-of-movement, as 3 Figs. 6, 9, of the complenient-form tool T, (Figs. 11, 12), relative to the position of the master-form axis :0, may be said to be a result of three features comprising, first, the c0ne-of-revolution of the master-form-axis of the wheel, as 2", Fig. 6; second, the angle subtended by the pitch-arc, as 7 Fig. 8; and, third, the angle, as 2 (Fig. 10), subtended by the profile lines of the master-form, F. With these several factors proportioned and arranged as in the accompanying drawings, the cone-of revolution, 2 of the masterform-axis 03, subtends a somewhat larger angle, than does the pitch-cone z' ,see Fig. 6. The said master-form angle, 2 and the length of the pitch-arc, result in locating said complement-form a.:is 3 in the described position so that the tool-edges 01 4) (Fig. 11) while the tool T, is moving along in parallel with said axis y, (Fig. 12), shall move, one tool-edge in touch with the surface of one tooth, and the other tool-edge in touch with the oppositely-disposed surface of an adjacent tooth. Thus by and while moving in parallelism with one axis, (21. e. an axis of the complement-form) the tool-edges generate surfaces which are also in another parallelism with the master-form axes, respectively.

Considering a pair of the adjacent masterform axes, as m, m, Fig 6 (also see Figs. 9, 9), as being straight-lines, and also assuming that the master-forms, as F, are geometric planes which as applied to the wheel-teeth, move along the respective axes while at rightangles thereto, the result will be the generation of a pair of geometric solids B and B arranged as illustrated in projection in Fig. 7. In this view while each of the profile lines, of each said form, extends to intersecting points at 8 and 6, the utilized master-form area,which may be said to constitute the master-form proper,-only extends in this instance from the outer line 8 (as shown in Fig. 7) to the inner depth- V parallelism of the master-form solid, (as applied to the wheel), and of the coincidence of the tapering dedendum surfaces of the wheel-teeth with the side surfaces of such solid, the wheel-tooth may be said to be eluded, or comprehended, within the geometric masterform solid, and to have throughout the entire-tooth-surface thereof, surfaceelements which are in parallel with each other and also with a master-form axis that is located within the said solid. It will also be evident that while this series of surface elements may be said all to extend, (as regards the master-form solid) the whole length of such geometric solid, these surfaceelements, as regards the tooth-surface itself, are of a decreasing length from the line at 6 (see Figs. 7, 9 and 9 downwardly there 7 from. This series of surface-elements, therefore, will terminate at the lower circle which defines the lower limit of said outer-end depth-zone.

It will be understood that as regards the parallelism of, the pair of outwardly-converging master-form surfaces, these surfaces, as to the utilized portions thereof, extend down through the depth-zone which indicates a proper lower limit for the working tooth surface of the wheel teeth, at the outer ends of these teeth. But since these wheelteeth, as shown in Figs. 7 and 9 are considerably inclined one relatively to the other as seen in plan view,see Fig. 8 .--the innerends of the master-form-solids B B are brought into an intersecting position, as indicated by the profile lines intersecting at the point 6 Thus the lower and inner portions of the master-form surfaces, become unavailable as regards tooth-faces in coincidence therewith; and also in the operation of finishing the tooth-faces by a counterpart tool corresponding to the outwardly-converging pair of master-form surfaces, this operation could not be extended to the entire tooth-surface, because such a tool, while finishing those surfaces on one tooth, and if extending through the full depth of the meshing Zone, would also cut away and so destroy the inner ends of the adjacent teeth, since the inner-end depth-zone is much less than said outer-end depth-zone.

When the profile lines and the form-axes are each a straight line, the pinion tooth faces may be said each to have a generated curved surface conforming to the compoundreproduction configuration developed according to a geometric plane-surface which, as related to the pinion, has a parallelism with a master-form axis that is a straight-line element of a geometric solid, or cone-of-revolution having a greater length and a smaller inclusive angle, or cone-angle, than the pinion pitch-solid, or pitch-cone; and which, as related to the wheel, has a parallelism with a complement-form axis that is a straight-line element of a geometric solidof-revolution,here shown as a cone-of-revolution,-having greater length and a smaller cone-angle than the wheel pitch-cone. These relations are indicated in Figs. 5 and 6, where the pinion master-form cone 2* is shown coinciding with the pitch-cone 2 at the outer-end pitch circle (see y, Fig. 5) while the apex thereof is shown extended to the point 0 well beyond the apex 0 of said pinion pitch-cone; therefore the coneangle of said master-form cone is a smaller angle than the corresponding angle 2 of the pitch-cone. Similarly, as best shown in Fig. 6, the wheel complement-form cone, .2, is shown coinciding with the pitch-cone 2 at the inner-end pitch circle 6 while the apex thereof is shown located far beyond the apex c of said wheel pitch-cone, so hat the cone-angle (as indicated by the are 2 of the said complement-form cone, is a smaller angle than the corresponding angle, 2 of said pitch-cone of the wheel; and, so that said cones e and 2 at the point-portions thereof, are geometric solids so located that the surface of each one cuts the surface of the other.

From the foregoing description as illustrated, it will now be evident that the whole series of tooth-faces are arranged in alternating pairs, of which the whole number of pairs is equal to the whole number of those faces. One series of pairs are outwardlyconverging, as and 7, while the intermediate pairs, as f and F, are inwardly-converging. (See Fig. 2). Also, the adjacent faces of the pairs of one series constitute the pairs of the other series, and each face of a pair in one series has an angular advance varying from that of the corresponding face in the pairs of the other series. (See Fig. 8). The faces comprised in the pairs in one series are arranged in the longitudinal parallelism and with one convergence, while the faces comprised in the pairs of the other series have an opposite convergence together with a longitudinal parallelism in a diiferent or varying direction. And in each set of three successive tooth-faces, there are two adjacent faces comprised in a pair in one said series, and two adjacent tooth-faces comprised in a pair having the inward-convergence. For instance, in Fig. 9 the two adjacent faces comprised in the pair h f and 7L are outwardly-converging and have a parallelism longitudinally of the masterform axis line 0, while the adjacent faces comprised in the pair h f and 7L f are inwardly-converging and have a parallelism longitudinally of the line, y, (Figs. 6 and 9) which is located in a direction diiferent from that of said axis line From the foregoing description, it will now be evident that an advantageous feature of the bevelgearing construction herein set forth is that the pinion tooth-faces and especially the addendum faces thereof,are tapering in width, while the tooth-faces are located and extend between converging dedendum and addendum cones, which may be located (one or both of them) with their apexes at the apex c of the geometric pitchsolid, or pitch-cone, and with a convergence of one or both of them (but preferably both of them) toward such intermediate pitchcone. Said cone-convergence is, of course, in the direction of the wheel axis, and the amount of the convergence is preferably such as will bring all three of these cones nearly as quite to a common apex. as c, as already explained in connection with Figs. and 6. Also, said pinion has each said tooth-face with a generated curved surface conforming to the compound-reproduction configuration developed according to a. geometric generation-surface (here shown as a plane) which.as related to the pinion,- has parallelism with a master-form axis, (a2, Fig. 6), that. in the plane of symmetry. (here indicated by line b", Fig. 9 is an element of a geometric solid. or cone-of-revolution, a, Fig. 6, having a less length and a larger cone-angle or inclusive angle than the cone, 2 Fig. 5; and which.-as related to the wheel,has a parallelism with a complement-form axis, as y, that is an element of a geometric-solid. or cone-of-revolution having a greater length and a smaller cone-angle, or inclusive angle, 2 (Fig. 6) than the wheel pitch-solid or cone 2 Fig. 6. And the comeshing wheel has the tooth-faces thereof located in coincidence with geometric surfaces which are in positions corresponding to the position of said generation-surface according to which the said pinion toothfaces are generated.

When the two profile angles, as f f Fig. 2 of the master-form, F, are equal, and if the master-form axis is a straight-line, a section of the wheel-tcoth taken in the plane of, or in any plane parallel to, the masterform axis :0, will be a parallelogram, while a section in the plane of a pitch-cone element, as 7 Fig. 6, will be a symmetrical quadrilateral of which said element is the axis of symmetry,-it being understood that said sectional planes are at right-angles to the axis of symmetry, as 5 Fig. 9, of the masterforin. -Tl11lS the-working surfaces of the teeth may be considered as being, or as being coincident with, outwardly-converging tooth-bounding surfaces which have straight-line profiles throughout the length thereof, and which are true planes when the master-form axis is a truly straight line.

The bevel wheel, 13, is shown provided with non-parallel teeth, and with each tooth located between a pair of outwardly-converging bounding planes which constitute working surfaces. Each pair of said working-surfaces comprises a pair of wheel-tooth addendum-surfaces arranged in one longitudinal parallelism, and also comprises wheeltooth dedendum-sin'faces each having an increasing width from the inner-end pitchcircle outwardly to the outer-end pitchcircle, while the adjacent dedendum-surfaces of adjacent wheel-teeth are arranged in another longitudinal-parallelism. For operating with the wheel, the pinion P is herein shown with teeth each having working surfaces comprising a dedendum portion curved to conjugationally conform with the said addendum-surfaces of the wheel-teeth, and each of these working-surfaces also comprises an addendum-face which is of an increasing width from said inner-end pitchcircle outwardly to said outer-end pitchcircle, and which is a conjugate geometrical envelop of the lines of contact of the addendum face of the pinion withsaid dederdum-face of the wheel-teeth.

In a pair of intermeshing conical gears, and as between the body of a wheeltooth and the body of a pinion-tooth, the proper rolling movement requires, in any given instance, some aggregate amount of transverse-curvature, or profile deviation, of the one tooth-face relatively to the other, and in the former practice it was customary to apportion that total relative curvature one part to the wheel and the remaining part to the pinion. In this system of conical gearing, and contrary to that former practice, no such apportionment necessary, and the whole of such relative curvature or deviation may be applied to the pinion-toothface, thus leaving the wheel tooth-face without any transverse-curvature. This feature combined with the longitudinal parallelism, and with the transverse convergence, is a means for bringing the two said members of the pair of gears into such a relationship that in addition to having improved opera tional features, the wheel-teeth can be made by the single-reproduction method, and only the pinion will require the compound-repro duction method for its manufacture.

In the usual forms of bevel-gear teeth,

the inner end of the tooth, since in such gearing all the tooth-face surface-element lines eX- tend to one point of origin, which in general practice (except in slrew-bevcls) is the point of axes-intersection. That uniformity. of ratio as between the trans -ersc tooth-section at any point in the length of the tapering tooth, is radically departed from in the pair of bevel-gears her in illustrated, in these gears, one of the wheels has a tooth-section of uniform character and of a variable section and-circle ratio, while in the mating wheel the teeth have both a variable toothsection and a variable said ratio as between such teeth-sections at successive points along the length of the teeth, and the corresponding pitch-cone circles.

In Figs. 11 to 16, l have illustrated one method of malrin the faces of the wheel,

and the use of the counter-part tools for making of the conjugationally-curved pinion-tooth faces by the duplicate of the master-form of the wheel with which such pinion is to mesh. In Fig. 11, the ellipse 8 is the outer circle of the wheel B, as seen in the direction shown in Fig. 12 the line of sight being parallel to the axis 3 This view also shows the plane-surface tooth-faces arranged outwardly-converging. In Fig. 12, a section is shown of the tooth-rim of the wheel B as seen from the ri 'ht-hand in Fi usual tool-holder of such well-known shap in g machine, the tool may then be fed gradually downward, thereby reproducing the form of the tool in a reversed arrangement in the'wheel inc and withfonly a' single direction or kind of movement as between the wheel and the tool, and thus completing one pair of plane-surface, parallel toot form faces by the "single-reproduction method. Tl operation being repeated around the wheel each pitch-arc interval, all the tooth-faces will be similarly cornpleted. Such a counterpart tool is shown at J, Fig. 13, as employed for making the pinion-tooth g 01"" a con}agate-curvature for correctly operating with the aforesaid toothform faces 7 and f oi the wheel. For this purpose, the tool 5 will be gii'en a planermovement with a rapid reciprocation in the line of the tooth-form axis and at the same time will be given a slow feed movement in a circle, as 8 (herein seen as an ellipse) coinciding with the pitch-cone or" the wheel; that is, the tool J hasa coinpoundsnotion, wa rep at d r i ent r e el essee ing movement in the line of the znasten form-axis of the p tch-cone oi the wheel, while this line of movement slowly revolves about the axis of the wheel. Thus the two counterpart tools produce :t'our faces arranged in two pairs for direct co-action, and arranged on the lines of master-forms which are identical for both the conical wheel and the conical pinion; and with the further feature that two of such four taces are also arranged on the lines of a complement-form which is not identical with the niastentorin.

When the master-form, as F, is located with its central line or axis w, in a plane radial to the wheel-axis and when at the same time the pair or" tooth-faces, as ,7, f outwardly from the innercircle 6 are longitudinally-converging relatively to the radii Z and Z respective y, as shown in F 8, then the .action face has a rearward deflection or skew-angle Z and the reaction t'ace f has a corresponding forward deflection or skew-angle. In practice (and except in shew-gearing) these forward and rearward tooth-face skew-angles are preferably equal and of such amplitude as to bring the tooth-face surface-elements into parallelism, since this construction and arrangement facilitates the manufacture and maintenance of the wheel teeth.

The pinion'tooth-surtace may be properly described as having a developee curvature generated according to the tooth-faces of the mating master-wheel of the par. The pinion tooth-faces are said to be generated because in practice they will he, usually and preferably, produced in and by some suitable gear-tooth-generating machine, the tooth-faces being developed from and by a too whose cutting lines are rnade according to the form of the master-wheel teeth, so the resultantor conjugate tooth-- face transverse curvature will be of the same developed formation as the toothfaces oi the pinion were actually molded into shape by the rolling 01" the pinion mesh with its own n'iaster-wheel. An in:- proved machine which i have devised for making the pinions in the znanner ere indicated, forms the subject-matter of separate application, Serial No. 814,035, filed January 24, 1914- for gear-cutting machine.

In the foregoing description I have set forth the manner in which the wheels and pinions, respectively, may be produced the processes of the single-reproduction and the coxnpourid-reproduction, and may be SO made by means of suitable machinery, but it should be understood that in some cases, if desired, these gears be reads-by casting them in suitable .niolds, by the use of gear-patterns previously prooucediby such machineniethods, or otherwise, and having the tooth-surfaces thereof, "ads 10f quired configuratio s.;-. -Also, jc-he L s nay be first cast as here explained, but with some surplus material on the working surfaces of the teeth, and these afterward reduced to exact size by means of the described mechanical operations.

Having thus described my invention, I claim:

1. In bevel gearing, a gear wheel having non-parallel teeth each located between a pair of bounding planes of a tapering width and forming the working surfaces thereof, and which are located with one longitudinal parallelism relatively to each other, and each of said planes also being located in another longitudinal parallelism with the adj acent plane working surface of an adjacent wheel tooth, in combination with a comeshing pinion having teeth provided with working surfaces of a tapering width and of a conjugate curvature conforming to both of said longitudinal parallelisms of said wheeltooth planes.

2. In bevel gearing, a bevel wheel having non-parallel teeth each located between a pair of outwardly-converging bounding surfaces constituting working surfaces and which have straight-line profiles throughout the length thereof, each pair of said working-surfaces comprising a pair of wheeltooth addendum-surfaces arranged in one longitudinal parallelism, and also comprising wheel-tooth dedendum-surfaces each having an increasing Width from the innerend pitch-circle outwardly to the outer-end pitch-circle, and said wheel also having the adjacent dedendum-surfaces of adjacent teeth arranged in another longitudinal-parallelism, in combination with a pinion provided with teeth having working surfaces each of which comprises a dedendum-face curved to conjugationally conform with the said addendum-surfaces of the wheel-teeth, and also comprises an addendum-face having an increasing width from said innerend pitch-circle outwardly to said outer-end pitch-circle, and which is a conjugate geometrical envelop of the lines of contact of the addendum-face. of the pinion with said dedendum-face of the wheel-teeth.

3. In bevel gearing, a bevel wheel having non-parallel teeth each located between a pair of outwardly-converging bounding planes constituting working surfaces, and having each pair of said working-surfaces comprising a pair of wheel-tooth addendum-surfaces arranged in one longitudinal parallelism, and also comprising wheel-tooth dedendum-surfaces each having an increasing width from the inner-end pitch-circle outwardly to the outer-end pitch-circle, said wheel also having the adjacent dedendumsurfaces of adjacent teeth arranged in another longitudinal-parallelism, in combination with a pinion provided with teeth havin" working surfaces each of which com" prises a dedendum-face curved to conjugationally conform with the said addendumsurfaces of the wheel-teeth, and also con1- prises an addendum-face which is of an increasing width from said inner-end pitchcircle outwardly to said outer-end pitchcircle, and which is a conjugate geometrical envelop of the lines of contact of the addendum face of the pinion with said dedendum-face of the wheel-teeth.

4. In a pair of master-form bevel-gears, a bevel pinion having the teeth thereof provided with tooth-faces extending outwardly from a dedendum cone to an addendum cone which in a direction toward the wheel-axis converges toward a geometric pitch-cone located between said dedendum and addendum cones, and also having each said tooth-face ofa generated curvature conforming to a compound -reproduction configuration developed according to a geometric generation-surface having straight-line profiles throughout the length thereof, and which (as related to the pinion) has one series of parallel surface elements in parallelism with a master-form axis that, when in the plane of symmetry, is an element of a geometric cone-of-revolution having a greater length and a smaller cone-angle than the pinion pitch-cone, and which (as related to the wheel) has another series of parallel surface elements in parallelism with a complementform axis that is an element of a geometric cone-of-revolution having a greater-length and a smaller cone-angle than the wheel pitch-cone, in combination with a comeshing bevel wheel having tooth-faces of the singlereproduction configuration and located in coincidence with geometric surfaces which are in positions corresponding to the position of said geometric generation-surface according to which the said pinion tooth-faces are generated.

5. In a pair of master-form bevel-gears, a bevel pinion having the teeth thereof provided with tooth-faces extending outwardly from a dedendum cone to an addendum cone which in a direction toward the wheel-axis converges toward a geometric pitch-cone located between said ded ndum and addendum cones, and also having each said toothface of a generated curvature conforming to a compound-reproduction configuration developed according to a geometric generation-plane which (as related to the pinion) has one series of parallel surface elements in parallelism with a master-form axis that, when in the plane of symmetry, is an element of a geometric cone-of-revolution having a greater length and a smaller coneangle than the pinion pitch-cone, and which (as related to the wheel) has another series of parallel surface elements in parallelism with a complement-form axis that is an ele ment of a gecmetric coneof-revclutien havpinion and a comeshing wheel, a bevel pinion having the tooth-faces thereof tapering in width and extending between converging dedendum and addendum cones located with the apex of one of them at the apex of the geometric pitch-solid of the pinion, and said pinion also having each said tooth-face of a generated curvature conforming to a compoundreproduction configuration developed according to a geometric gene'ration-surface having straightline profiles throughout the length thereof, and which (as related to the pinion) has one series of parallel surface-elements in parallelism with a mastenform axis that, when in the plane of symmetry, is an element of a geometric solid-of-revolution having a greater length and a smaller inclusive angle than the pinion pitch-solid, and which (as related to the wheel) has another series of parallel surface -elements in parallelism with a complement-form axis that is an element of a geometric solid-of-revolution having av greater length and a smaller inclusive-angle than the wheel pitch-solid, in combination with a comeshing wheel having tooth-faces of the single-reproduction configuration and located in coincidence with geometric surfaces which are in positions corresponding to the position of said geometric generation-surface according to which said pinion tooth-faces are generated.

7. In a pair of bevel-gears comprising a pinion and a comeshing wheel, a pinion having the tooth-faces thereof tapering in width and extending between converging dedendum and addendum cones each located with the apex thereof at the apex of the geometric pitch-cone of the pinion, and said pinion also having each said tootlrface of a generated curvature conforming to a compound reproduction configuration developed according to a geometric generation-plane which (as related to the pinion) has one series of parallel surface-elements in parallelism with a master-form axis that, when in the plane of symmetry, is an element of a geometric cone-of-revolution'having 2 g i ater length and a smaller inclusive angle than the pinion pitch-cone, and which (as related to the wheel) has another, series of parallel surface-elements in parallelism with aeoInplement-form axis that is an element of fa'fgeometric eonepf-revelution having a greater length and a smaller coneangle than the wheel pitch-cone, 1n combination with a comeshing bevel-wheel having the tooth-faces thereof of the smgle-reproduction configuration and located in coincidence with geometric planes which are in positions corresponding to the position of said geometric generation-plane according to which said pinion tooth-faces are generated. V i

8. In a pair of master-form bevel-gears, a pinion having the teeth thereof provided with addendum-faces tapering in width and extending outwardly from the geometric pitch-solid to an addendum-cone which, in a direction toward the wheel-axis, converges toward said pitch-solid, and also having each said addendum face of a generated curvature conforming to a compound-reproduction configuration developed according to a geometric generation-surface having straight-line profiles throughout the length thereof, and which (as related to the pinion) has one series of parallel surfaceelements in parallelism with a master-form axis that, when in the plane of symmetry, is an element of a geometric solid-of-revolution having a greater length anda smaller inclusive angle than the pinion pitch-solid, and which (as related to the wheel) has an other series of parallel surface-elements in parallelism with a complement-form axis that is element of a geometric solid-ofrevolution having a greater length and a smaller inclusive angle than the wheel pitch-solid, in combination with a bevelwheel having pinion-engaging tooth faces of the single-reproduction configuration and arranged for comeshing with said addendum-faces of the pinion, and also conformin profile and position with the profile and position of said geometric generation- ,surface according to which the addendum faces of the pinion are generated.

9. In a pair'of master-form bevel-gears comprising'a conical pinion and a comeshing wheel, a pinion having the teeth thereof provided with addendum-faces tapering in width and extending outwardly from the geometric pitch-solid to an addendum-cone which in a direction toward the wheel-axis converges toward saidpitch-solid, and also having each said addendum face of a generated curvature conforming to a compound-reproduction configuration developed according to a geometric generationplane which (as related to the pinion) has one series of parallel surface-elementsin parallelism with a master-form axis that, when in the plane of symmetry, is an element of a geometric solid of-revolution haviag a greater length and a smaller'inclusive angle than the pinion pitch-solid, and which (as related to the wheel) has another series of parallel surface-elements in parallelism with a complement-form axis that is an element of a geometric solicl-ofrevolution having a greater length and a smaller inclusive angle than the wheel pitch-solid, in combination with a bevel wheel having pinionengaging tooth-faces of the single-reproduction configuration and arranged for comeshing with said addendum-faces of the pinion, and also conforming in profile and position with the profile and position of said geometric generation-plane according to which the addendum faces of the pinion are generated.

10. In a pair of master-form bevel-gears comprising a pinion and comeshing wheel, a conical pinion having the teeth thereof provided with addendum-faces tapering in width and extending outwardly from the geometric pitclrcone to an addendum-cone which in a direction toward the wheel-axis converges toward said pitch-one and also having each said addendum face of a generated curvature conforming to a compoundreproduction configuration developed according to a geometric generation-surface having straight-line profiles throughout the length thereof, and which (as related to the pinion) has one series of parallel surfaceelements in parallelism with a master-form axis that, when in the plane of symmetry, is an element of a geometric cone-of-revolution having a greater length and a smaller inclusive angle than the pinion pitch-cone, and which (as related to the wheel) has another series of parallel surface-elements in parallelism with a complement-form axis that is an element of geometric cone-ofrevolution having a greater length and a smaller inclusive angle than the wheel pitch cone, in combination with a conical wheel having pinion-engaging tooth-faces of the single-reproduction configuration and arranged for comeshing with said addendumfaces of the pinion, and also conforming in profile and position with the profile and position of said geometric generation-surface according to which the addendum faces of the pinion are generated.

11. In a pair of master-form bevel-gears comprising a pinion and a comeshing wheel, a conical pinion having the teeth thereof provided with addendum-faces tapering in width and extending outwardly from the geometric pitch-cone to an addendum-cone which in a direction toward the wheel-axis converges to an apex which is coincident with the apex of said pitch-cone, and also having each said addendum face of a generated curvature conforming to a compoundreproduction configuration developed according to a geometric generation-plane which (as related to the pinion) has one series of parallel surface-elements in parallelism with a master-form axis that, when in the plane of symmetry, is an element of a geometric cone-of-revolution having a greater length and a smaller inclusive angle than the pinion pitch-cone, and which (as related to the wheel) has another series of parallel surface-elements in parallelism with a complement-form axis that is an element of a geometric cone-of-revolution having a greater length and a smaller inclusive angle than the wheel pitch-cone, in combination with a conical wheel having pinion-engaging tooth-faces of the single-reproduction configuration and arranged for comeshing with said addendum-faces of the pinion, and also conforming in profile and position with the profile and position of said geometric generation plane according to which the addendum faces of the pinion are generated.

HARVEY D. WILLIAMS.

Witnesses."

H. D. PENNEY, FRED. J. Donn.

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