194 MR E. SANG ON THE TOOTHING OF UN-ROUND DISCS. 
obliquely, wherefore the general character of its path must be as represented 
in figure 5; and, when there is only 
one contact-point, that point must 
trace the curve QMGK FQILHQ 
continuously during the passage of 
each tooth. In order that this be 
possible, the distance @P must 
increase while P moves along the 
part QGK, and must attain its 
maximum at K. Whenever P 
passes into the part K FQ, QP 
Fig, 5. must decrease because the cosine 
of RQ P is then subtractive ; and similarly for the remaining half of the path. 
Teeth formed by help of such a path would be useless in determining 
mechanically the relative positions of two discs; the single contact might 
prevent A from moving forward without taking B along with it, but then it 
would not hinder A from being turned backwards ; wherefore it is requisite to 
have more contacts than one. 
If the contact-path pass outside of the circle described from Q with the 
radius Q K, as in figure 6, the motions at 
K and L must be upwards, or in the same 
direction with that of the pitch-lines. Let 
F, G, H, I, be the four points at the maxi- 
mum distance from Q, then if the contact 
begin at F, it must separate there into 
two, one contact proceeding along F Q L 
the other along F K G. The former of 
these will reach I just as a third contact 
proceeding along the line H L I arrives 
there, and the two coalescing will cease. 
In the same way that contact which has 
passed along F K G will coalesce at G with another that has passed along 
HQG:; and the motions will be timed so as that the arrivals at K, Q, L, Q, 
divide the entire passage of a tooth into four equal parts. 
Since the contacts appear and disappear in pairs, their number must be 
odd, excepting just at the instant of beginning or ending. By augmenting the 
ordinates parallel to RS, thereby making the curve flatter at K and L, we can 
increase the number of contacts ; and it is possible so to determine this aug- 
mentation as that a new contact may appear at F or H just as a disappearance 
occurs at Gor I. When this has been. done, the number of points in contact 
remains constant, and these are distributed so that the total number of them on 
R 


Fig. 6. 
C—O 
