MR E. SANG ON THE TOOTHING OF UN-ROUND DISCS. | 195 
the two parts FQ I, HQG, exceeds the total number on FK G, H LI, by | 
one. . 
The simplest arrangement admissible in practice is that with three contacts. 
Of these one, occuring on H QG, is on the front of a tooth, a second occuring 
on F QI is on the back, while the third happening alternately on F K G and 
H LIis on the top or bottom of the tooth. The first and second of the 
contacts serve to determine the relative positions of the discs in the direction 
of their pitch-lines, and to communicate the motion of rotation from the one to 
the other. The third serves to resist the pressure exerted to keep the discs in 
contact. 
When the discs turn on fixed centres the resistance perpendicularly to their 
pitch-lines is exerted on the centre pins or axes; but, in the case of un-round 
discs generally, the action at the tops and hollows is indispensable. 
Tf such a contact-path graduated for equidifferent positions of the pitch- 
lines, be carried round any of the discs A, B, C, in such a way as that the point 
Q is placed at the successive points of division while the line A QB lies along 
the normals at these points, marks made through its appropriate divisions upon 
the plane of the disc, ‘indicate the forms of the teeth ; and the discs so toothed 
work truly with each other. This mode of delineating the teeth of irregular 
discs is a generalisatign of that explained in my “ New General Theory of Wheel 
Teeth ” as applicable to ordinary wheels. 
The teeth placed round any one of these discs necessarily vary in shape accord- 
ing to the radius of “curvature of the pitch-line ; their mechanical possibility is 
limited by certain considerations. If, while a point moves along the part Q G, 
a normal to the curve accompany it, the point at which that normal crosses 
Q B, will move away from Q, will reach a maximum distance and thence return 
to Q. Whenever the centre of curvature of the pitch-line lies within this 
extreme distance, the outline of the side of the tooth becomes folded in; also 
whenever it lies within the extreme limit of the normal applied to the curve on 
the part LI, the outline is replicated at the top of the tooth ; and the greater 
of these two limits indicates the impossibility of the mechanical action. This 
matter is thoroughly discussed in a paper entitled “Search for the Optimum 
System of Wheel-Teeth.” 
VOL, XXVIII. PART I. 3E 
