BEVIt GEAR.] 



APPLIED MECHANICS. 



891 



the pitch circle, half the space E F between two teeth, 

 and A K the depth of the space below, greater than A W 

 by Jth of an inch for every inch of pitch ; and circles 



Fig. 242. 



of a tooth at K is part of a circle described with radius 

 K H. The tops of the teeth in this case project loithin 

 the pitch circle, as far as in external gearing they project 

 beyond it, and the bottoms of the spaces may 

 be made semicircular, and of depth similar 

 to that determined for external gearing. 



The gearing we have hitherto described 

 applies only in the case of wheels revolving 

 in one plane or on parallel axes. 



BEVIL GEAR. When the axes are not 

 parallel, it is necessary to employ beril gear. 

 Let A B and A C (Fig. 243) be the two 

 axes meeting in A, and D E and E F the 

 proper diameters of the wheels to give the 

 required speed ; if we suppose the cones 

 DAE, F A E to roll upon one another 

 touching along the line A E, then the velo- 

 cities of their touching surfaces will be equal 

 at any point, such as M, along the cone, 

 because L M and K M, the radii of rotation 

 at that point, bear the same proportion to 

 each other as G E and H E the radii at any 

 other point E, and therefore the circum- 

 ferences at these points are in like propor- 

 tion. Taking a portion of each cone (that 

 shaded), if we conceive their meeting sur- 

 faces at M E to have sufficient friction, by 

 causing one to rotate round its axis, we 

 should also cause the other to rotate. But 

 as practically the friction is not sufficient, it 

 is necessary to cut the surfaces of the cones 

 into teeth and spaces, as in plain gearing. 

 It is evident that these teeth must taper 

 towards A, both in width and in height. If 

 a line C E B be perpendicular to A E, and 

 N and P the tops of the teeth at E, the con- 

 verging lines A N and A P will define the 

 tops of the teeth along their whole extent. 

 tig. 243. 



described through W and X will determine the tops of 

 teeth and bottoms of spaces. 



In the case of internal gearing (as represented in Fig. 

 242), the form of teeth may be developed on principles 

 similar to those we have adopted for external gearing. 

 If A B (Fig. 242) be the pitch radius of the 

 wheel, F B G being a portion of the pitch 

 circle, and C B the pitch radius of the 

 pinion working withiu it, and if on the 

 axes of the wheel and pinion we suppose 

 pulleys to be fixed, having radii A D and 

 C E, respectively proportional to the pitch 

 radii, a band E D winding off the one and 

 on to the other pulley would give them the 

 same relative angular velocities as would 

 be produced by the friction of their cir- 

 cumferences at B. And D E, the common 

 tangent of the two pulleys, which, when 

 prolonged, must always pass through B, 

 may be supposed to have a pencil fixed 

 to its prolonged part, tracing on the planes 

 of the revolving wheel and pinion, the 

 outlines of their respective teeth, which 

 would manifestly be involutes of the gene- 

 rating circles D and E. 



The inclination of D B being the same 

 as that adapted for external gearing, the 

 teeth of the pinion would evidently be the 

 same, but the teeth of the wheel would be 

 similar in form, not to the teeth of an ex- 

 ternally geared wheel, but to the spaces between them. 

 L being the centre of the wheel, LH = 3^nds of the 

 pitch radius, and H K= Jth of the pitch radius, the side 



And again, if Q B be the breadth of a tooth at H, the 

 converging lines A Q and A B will define the tapering 

 breadth. The Hue P N is part of the boundary of coni- 



