610 
Proceedings of the Royal Society 
a 
Let two circles touch one an¬ 
other at their lowest points— 
compare the arcual motions of 
points P and p, which are always 
in the same horizontal line. 
Draw the horizontal tangent 
AB. Then, if the line MPp be 
slightly displaced, 
AM ' 
continuously 
fflM . MO 
Arc at P AO Mp AO / 
Arc at p ~ MP ’ dO “ oO > AM . MO 
AO / 
“ a(W 
Thus, if P move, with velocity due to g and level a, 
in its circle, p oscillates with velocity due to 
a 
0 J 
AO 5 
and level AB . 
Combining the two propositions, we are enabled to compare the 
times of oscillation in two different arcs of the same or of different 
circles. 
Professor Cayley has pointed out to me that results of this kind 
depend upon one of the well-known fundamental transformations 
of elliptic functions. In fact, it is obvious that the squares of the 
sines of the quarter arcs of vibration which the combination of the 
above processes leads us to compare are (in the first figure), 
CA C'B 
CE and C D 
respectively* 
Calling them 
and putting DA 
1 _ e 1 2 s /2ae + e 2 
k 2 ~ 2a + e ’ k» ~ a + e + J2ae + e 3 ' 
we have 
