358 
Proceedings of the Royal Society 
for which there must be seven contacts, and the first stationary posi- 
tion of N must be brought forward to V, as is the case in fig. 8. 
The most convenient process for tracing the shapes of wheel- 
teeth from such a rack is to determine the positions of the contact 
points corresponding to equidifferent positions of the wheels, and 
to combine the motion along the path with the proper angular 
motion of a blank disc. In this way we obtain very readily the 
outline of the wheel ; this outline, however, though always giving 
the proper number of contacts geometrically, is not always mechani- 
cally possible ; for low-numbered wheels it is exceedingly convo- 
luted, as is seen in fig. 9, which is that of a wheel of one tooth, 
belonging to the system of fig. 8. As the number of teeth is 
augmented, the convolutions become less, and at a certain limit, 
the limit of mechanical possibility, they disappear. 
Hence arises an exceedingly important and most difficult problem, 
“ To discover that form of rack which, giving a determinate number 
of contacts, shall admit of the lowest numbered wheels.” The idea 
of the curve of second sines occurred in the attempt to resolve this 
problem. 
Here the condition of optimism cannot be put in the form of 
maximum or minimum, so that the known methods of analysis are 
inapplicable, and we must have recourse to successive trials with 
known or with invented lines, and after all we can only conclude that 
such or such a curve is preferable to any other that has been tried. 
The simplest line, consisting of an endless series of equal and 
symmetric undulations, is the well-known curve of sines ; this 
curve, when used as the form for a rack, gives convolutions on the 
outlines of wheels of considerable size, and it becomes desirable to 
obtain a better shape ; for this we are naturally led to try modifi- 
cations of the curve of sines. 
If we write u for the absciss, y for the ordinate, and v for some 
variable arc, the equations 
u - v + <£ sin 2 v ; y - A . 0 sin v , 
in which </> and 0 represent two unknown functions, will give equi- 
distant and symmetric undulations, it being essential, however, 
for our purpose that 0 = 0 ; #0 = 0 ; <£( — 2) = - <p( + 2) and 
0(_ 2) = ~0( + 2). We have thus an endless variety of modifi- 
cations among which to make our trials. 
