INVOLVING THE PRODUCT OE TWO LAPLACE’S COEFFICIENTS. 
581 
These results*' are immediate consequences of what is known with respect to the values 
of the integrals 
J +i 
in which the integration extends over the whole sphere ; for if n, n' are both odd or both 
even, is an even function of and so 
j Q„'Ch, (!(*—{ QnQn’df J j. 
The peculiar character of the integrals over the hemisphere only shows itself when 
one of the quantities n, n' is even and the other odd. 
The coefficient of e° in the expansion is 1 
coefficient of fl±^!=l . 
sxx • c 4 d 4 , 1 . 3 (1 +e n )% 5 l(l+e ,2 )i , l(l + e ,2 )§-l 
coefficient of ; 
coefficient of e 6 =~- 
X O 
gia 
coefficient of e 8 =— • 
17 
1 .3.5(l + e ,2 )i . 5.7 l(l+e ,2 )t 9 l(l+e ,2 )l, 1 (l + e ,2 )¥-l . 
2 3 (3 e 1 ‘ 2 2 |2 " 5 e 13 2 '9 e' 5 ‘ 13 P 7 ’ 
e 1 ■ 2 2 1 2 5 e 13 2 9 e 
1 .3.5.y(l+e f2 )y 5.7.9 l(l+e' 2 )fi 
2 4 |4 
2 3 (3 '5 
, 9.11 1 (l + e ,2 )& _13 1 (l + e' 2 )¥, 1 (l+e ,2 )ir-l 
9 e' 5 2 ’ 13 d 7 ~^"l7 e 19 
The law of formation of these series is obvious, and the coefficient of e 2n could, if 
necesssary, be written down. 
From the symmetry of the original expression in e and e } we know that the coefficient 
of e n e' n ' must be the same as that of e n 'e !n ; so that, in order to obtain all the integrals 
required, it is not absolutely necessary to consider the coefficients of odd powers of e. 
Nevertheless, in the calculation for instance of J QjQ 10 f7^, it would be much easier to 
obtain it as the coefficient of e no in the series which multiplies e, than as the coefficient 
of e' in the series which multiplies e 10 . 
The coefficient of e 
__e'_!_l (l+e ,2 )i-l. 
~ 3^~3 e' 2 
coefficient of e 3 
e' 3 3 1 (l + e ,2 )l , 1 (l+e ,a )i—l. 
•2 3 
e' 4 
* They would, of course, be more simply obtained by taking the integration in the first instance from p = — 1 
to /f=+l.— (Nov. 1870.) 
4 K 2 
