of increase of QP shall be to the 
lines of the teeth. In this way the 
MR E. SANG ON THE TOOTHING OF UN-ROUND DISCS. 195 
The sides of the trigon s7P are perpendicular, respectively to those of 
P Q a, wherefore 
Athias eas - 7: A@ es: Px 
Now Pas isthe angular displacement of the disc A, wherefore, if we make 
Qq parallel and equal to Pv, Q aq must also represent the same angular dis- 
placement. Also for the same reason P 6¢, and its like Q04q, represents the 
angular displacement of the disc B; so that the angular movements are in 
inverse proportion to the radii aQ and 6Q; in other words Q is the pitch- 
point and Q g or Pr the motion of the pitch-lines. 
During this minute change of position, the point of contact has moved from 
P to some point in the line s¢; the direction of its motion being undetermined. 
_ This investigation shows us, in the first place, that the line joining the pitch- 
point with the point of contact is always normal to the surfaces of the teeth. 
Secondly, since the augmentation of @P is the distance between the two 
parallels S T, sz, it follows that the velocity of the pitch-lines is to the rate 
of increase of QP as radius is to the cosine of the inclination ¢ QP. 
And thirdly, the motion of the contact-point is independent of the radii of 
curvature a Q, 6 Q, of the pitch-lines. 
Hence we have this noteworthy porism: If the point P move along some 
line, which we may call the contact- R 
path, in such a way as that the rate | p 
rate of motion of the pitch-lines, as 
the cosine of the angle R Q P isto the 
radius, the motion of P combined ° 
with the appropriate motion of any of 
the discs A, B, C, will generate, on 
the planes of those discs, the out- 

A 
Q 
Fig. 4. 
peculiar form of the pitch-line of the disc is eliminated, as it were, from our 
investigation, and we have to consider chiefly the form of the contact-path. 
Since the same action has to be repeated, the entire contact-path must be 
retraced during the passage of each successive tooth ; and therefore, if there be 
only one point of contact, the path must return into itself. Also, since the tooth 
outline must cross the pitch-line of the disc in all actual cases, the contact-path 
must pass through the point Q, and must lie partly on either side of the line 
Qq; it must also lie above and below A Q B, and so must be distributed among 
the four quadrants round Q, while it is obvious that for all business purposes, 
the four parts should be symmetrically placed. 
From the general law of its motion it follows that the point P cannot cross 
A QB obliquely except at Q, and that it cannot cross RQS otherwise than 
