530 
MR. HENNESSY’S RESEARCHES IN TERRESTRIAL PHYSICS. 
sions, and then the integrals may be tahen between the limits proper for each parti- 
cular case. 
Let X, y and ^ be transformed into polar coordinates as in article 2, and the sums 
of their squares taken two and two be put in such a form as to satisfy the equation 
of Laplace’s coefficients. We shall then have, putting (m for cos 6, 
Jl [l~ sin'o^J ^drdadi/j, 
^+1 ,,r r2 rl n't 
A, = 1^3— ( 1 cos^o^J 
But 
and 
f\r‘dr=\f^ g,^da=l f‘g~[a>(l+l3^)]da, 
W=W.+W,+W,+ ....+W,. 
From article 5 Wi=0, except when i=2 or 1 ; and in the latter case it is made to 
disappear when a' is the radius of a sphere equal in volume to the spheroid ; we have, 
therefore, on making 
d(a^) 
and 
Ja^^W^da 1 
J'jr*dr=\x{a!) + - 5m [f{o!)J^^ ^s^da-\f^ 
by article 5. Consequently 
Mm/ 
C,= ^ [y %, (a,) - +mal'l'{a , , a')] , 
(30.) 
where 
Hence 
^“1 
%.(«i)= / f 
c/ 0 
c=|]s- 
d{a^)da 
da 
3 j> 
(31.) 
making for brevity 
S=y[%(«')-%(«i)+%i(«i)5 (32.) 
'P = ei^iOj— «>i] • • (33.) 
