WITH THE TIDES OE A VISCOUS SPHEROID. 
563 
X % ' cos i 2i + 1 )x d X=2i^l JC j sin ( 2i + C0S ( 2 * + l ^ X ~j2 i + lf Shl ^ 2i '" lL 
d x_ 
+ - 
d x 
d Y j 
— cos ( 2 ^+ l)x-j-&c. 
d}_ 
dm* 
dA_ 
dm' 1 
(—V 
4 1 - +—^-& C . 
2% + lL (2i+l) 3 (2 j + l) 4 
(2i + 1) 3 
If therefore /(x) be a function of x involving only even powers of x> 
77 
f/(x) cos ( 2i + l h (l x=4 } 
(2t +1) 
i+i— y 
~ \2i + l dm 
A™)- 
This theorem will make the calculation of the coefficients very easy, for we have 
at once 
---—;[-2 L + 4 .3 M ' ra 3 -0.5NV>l 
+ ( 2 i+lh 4 - 3 - 2 ' 1 ' M '- 6 - 5 - 4 ' 3NV ] 
-( 2 iTlj' 6 [_ 6 ' 5 ' 4 ' 3 ' 2 'lN' ] |- 
Substituting for R/, L, M', N' their values in terms of 2 we find 
A/ 
_ (-)«. 2 a« fj 
2,+1 (2i + l)V k 
19- 
1988 
0216 
(2 i +1) 3 sj 2 (2 i + l) 4 ts 4 _ 
Then putting for — its value, viz.: ^ of 3T4159, and putting i successively equal to 
0 , 1 , 2 , it will be found that 
P,=*£(1201»07), P 8 =^(l-107), P 5 =- 6 -f(-048). 
So that the Fourier expansion is 
7 TX _ 07 TX „ , „ 07 TX 
120 907 cos -—j-1‘107 cos —— — ‘048 cos —, 
2 a Act Act 
which will be found to differ by not so much as one per cent, from the function 
tO| 3 
