192 MR E. SANG ON THE TOOTHING OF UN-ROUND DISCS. » 
T comes to the pitch-point as it is called, the normal to the curve at T must lie 
along the straight line S A, while the corresponding normal at V must be along 
SB; and the motion of each disc becomes an instantaneous rotation round the 
centre of curvature. 
Were such discs employed for the transmission of pressure, the limit of their 
action would be determined by the co-efficient of friction between the two 
substances, and would, at all times, be precarious. For the purpose of pre- 
venting any slipping of the one boundary past the other, we propose to notch 
and tooth them. The object of the present paper is to PGS SU the principles 
according to which this toothing must be accomplished. 
In the first place we oqserve that if the discs be to turn round and round 
upon each other, their peripheries must be commensurable, and the distance 
from tooth to tooth must be a common measure of them. 
We may next observe that when both discs are convex all round, they 
may roll upon each other ; but that if one of them have a sinuosity as in figure 
2, no part of the other’s contour can apply to the 
hollow unless it be convex and have its radius of 
curvature less than the osculating radius of the 
concavity. This, and similar limitations, caused by 
C the impenetrability of the solids, may be set aside 
when we are considering only the geometry of the 
subject. 
The contour of the toothed disc must undulate 
on either side of the pitch-line ; our first business 
ae is to inquire into the general law of its formation. 
Let a and 4, figure 3, be the centres of instantaneous revolution of two discs ; 
let P be the point at which 
their teeth touch, and let S PT 
be the direction of their com- 
mon tangent. 
move forward by an indefi- 
nitely small quantity, and that 
the common tangent takes up 
the new position sz, sensibly 
x parallel to ST; then making 
b q AaPs and b Pt each a right 
Fig. 8. angle, s and ¢ will indicate the 
new positions taken up by the points P of the discs, and s ¢ will represent the 
distance through which the peripheries slide upon each other. Draw Pr 
perpendicular to a 6, and P Q perpendicular to S T. 

* bi BF oa 
a 
_ 
Imagine now that the discs 
