[634 
635] 
17 
635. 
riTIONS. 
NOTE ON THE DEMONSTRATION OF CLAIRAUT’S THEOREM. 
p. 188.] 
attended to; e.g. 
series is 
[From the Messenger of Mathematics, vol. V. (1876), pp. 166, 167.] 
It seems worth while to indicate what the leading steps of the demonstration are. 
The potential of the Earth’s mass upon an external or superficial point is taken 
to be 
T r Fo Fi Vo 0 
F> - — + ~r + ^ + &c -> 
where V x , V 2 , V 3 ,... are Laplace’s functions of the angular coordinates. 
The surface is assumed to be a nearly spherical surface r = a (1 + u), where 
u = u x + u 2 + &c., and u x , u 2 ,... are Laplace’s functions of the angular coordinates. To be 
a surface of equilibrium, with an equation V+ ^&> 2 r 2 sin 2 6 = G, the latter must be 
equivalent to the equation r = a{\ + w), and it follows that we have 
V x = V 0 a u ly 
V 2 = V 0 a?u 2 — ^co 2 a 5 (| — cos 2 6), 
V 3 = V 0 a?u 3 , 
&c., 
which values are to be substituted in the expression for V. 
The whole force of gravity (due to the attraction and the centrifugal force) is 
taken to be g, — — (V + ^a> 2 r 2 sin 2 6), and it follows that 
CLT 
g = (1 + u 2 + 2u 3 + ...) - f ® 2 a — f co 2 a (£ - cos 2 6), 
CL