of Edinburgh, Session 1873-74. 
359 
On omitting the term cj> sin 2v, and putting 0sin v = sin sin?;, we 
obtain the equation 
y - A . sin 2 u , 
that of the curve of second sines. The transition from the ordi- 
nary wave-line to this curve is abrupt, and symbolically of the same 
nature as the transition from the straight line to the curve of sines 
itself, as is seen on comparing the three equations 
y = A . u ; y = A . sin u; y = A . sin sin u ; 
but for the elucidation of the theory of wheel-teeth we require a 
gradual transition from the one kind of curve to the other ; that is 
to say, we must obtain some comprehensive genesis which shall 
include both species of curve, and permit of an imperceptible 
change from the one to the other. 
If a body vibrate along the straight line AOB, in virtue of some 
elastic arrangement whereof the 
redressing tendency is propor- ? $ 5 A 
tional to the distance from the 
mean point 0, and if, while it is 1 ‘ 
so vibrating, a sheet of paper be carried over it with a uniform 
velocity in a direction perpendicular to AOB, the trace made on 
that sheet is a curve of sines. 
Instead of the rectilineal oscillation, let us use the motion of the 
balance-wheel of a watch — that is to say, let the vibration be in 
the circular arc AOB ; and while the abscissae, measured along the 
line BST, are made proportional to the times, let the ordinates be 
made equal to the sines pp of the arcs, instead of to the arcs Op 
themselves, and we shall have a variety of the curve of second sines. 
If the extent of the arc be small, its sinepp hardly differs from 
itself, and the curve merges 
into the ordinary curve of sines. 
When the length of the half- 
arc OA is just equal to the 
radius CO of the circle, we 
obtain the curve of second 
sines proper, which is repre- 
sented in figure 1, the base 
RV being equal to the circum- 
ference of the circle, whose radius is CO. As the extent of 
