356 Proceedings of the Royal Society 
there would be a maximum resistance with a certain degree of 
curvature This, however, cannot be proved by the small number 
of observations made curved surfaces, but would be very interesting 
to ascertain experimentally. 
2. On the Curve of Second Sines and its Variations. 
By Edward Sang, Esq. 
The idea of this class of curves arose during an attempt to 
resolve an important problem in the doctrine of wheel- work ; a 
statement of the conditions of that problem is thus the natural 
introduction to the subject. 
When the shape of the tooth of one wheel A has been assumed, 
the shape of the tooth of another wheel B, to work along with it, 
may he deduced by a very simple graphic process ; and when these 
two wheels are made to turn together, the points of contact describe 
a certain line or path. In my “ New G-eneral Theory of the Teeth 
of Wheels” it is shown that this path, and the manner of motion in 
it, are independent of the size of the second wheel B, and result 
entirely from the arbitrarily assumed form of A ; that is to say, if 
we delineate the form of a new wheel B, to work along with A, the 
path and the motion of the contact point along it are the same as 
before ; the originally assumed form A thus gives rise to a system of 
conjugate forms B. Not only so, but if we use any one of the 
wheels B as an originator, we shall obtain from it a system of con- 
jugates A, of which our first wheel A is a member. Thus the 
assumption of one wheel gives rise to two conjugate systems, A and 
B, so related that any wheel of the one works with any wheel of the 
other, the contact path remaining the same for every couple. It 
does not, however, follow that two wheels of the same set can work 
together; in the arrangement of wheel- work it is important that 
the two systems he identic. Now, when the wheel is indefinitely 
enlarged, its boundary merges into a straight rack, and the rack A 
is necessarily a copy of the rack B ; hence we come to the most 
important theorem in the doctrine of engrainage, that “ if we 
assume for the outline of a straight rack any curve consisting of 
equal undulations, symmetrically arranged on either side of its 
pitch-line, all the wheels determined by it work with each other, 
