460 Proceedings of the Royal Society 
or, if we note that 
V.a Via — a(ai — a sin A), 
( — x — 2ya>sin \ — ri*x)aYia -j- (ij — 2xa) sin \ + n % y)a(ai — a sin A) = 0. 
This gives at once 
x + n-x + 2 (nij sin A = 0 
y + n?y - 2a) x sin A = 0 
which are the equations usually obtained ; and of which the solu- 
tion is as follows : — 
If we transform to a set of axes revolving in the horizontal 
plane at the point of suspension, the direction of motion being 
from the positive (northward) axis of x to the positive (eastward) 
axis of y, with angular velocity O, so that 
x = £cosfi£ - r] sin 
y — $ sin tit + 7] cos 
and omit the terms in H 2 and in wO (a process justified by the 
results, see equation (15)), we have 
(£ + n 2 £) cos Ql - (ij + n 2 7j) sin Qt - 2 y(fl - w sin A) = 0 ) 
(i + sin Qt+ (tj + n 2 7] ) cos Clt + 2^(0 — w sin A) = 0 | 
So that, if we put 
0= (o sin A (15), 
we have simply 
the usual equations of elliptic motion about a centre of force in the 
centre of the Ellipse. 
5 . In the paper this problem is treated with a closer approxima- 
tion, terms in w 2 , &c., being retained. The conical pendulum — the 
path of the bob being very nearly a horizontal circle, i.e., the tension 
of the cord being nearly constant — is next treated ; then the case 
of very great angular velocity, when the path is nearly a circle 
(in any plane) with centre at the point of suspension. A few sec- 
tions are devoted to the consideration of the effect of a disturbing 
body, such as the moon or the sun. 
£ + n 2 £ = 0 
irj + n 2 7] - 0 
( 16 ), 
