2-20 
ME. S. P. HOUGH ON THE APPLICATION OP HARMONIC 
n .7T> 
_ 4a2a;2 (1 - /x2) S , : - ij}) 
^ ‘ ' n {n + 1 ) ' dfjb 
(21). 
But, by well known properties of the zonal harmonics, we have 
— n{n \ P„ f//r 
/n 2\ _ 
n(n + 1) I cl? 
2n +1 j \ dfi 
(l?n-, 
d jjb 
_ ('!'>■ + 1) /p _ p \ 
” 2?i + 1 1 
no firbitrary constant being necessary since both sides vanish when jx — 1 ; and 
therefore 
r/P 
(/' — ?') 
d 
ip 
{p 
whence 
f?P 
/ 2 \ ” 
‘'f 
p- 
n (n -|- 1) 
2a -h 1 
(P« + l 
- P.-l) 
p- 
a (a -1- 1) 1 
■ 1 
/ dP„+2 
d?n\ 
1 /dP„ 
dP„. 
2/1 + 1 ] 
,2 a -f 
3^, d^x 
d^j 
2n -\\dix 
dfx 
a (n -}- 1) 
d?n+o 
{2ii + l)(2?r + 3) d/j, 
2n(n + 1) 
+ U-i + rr 
rfp„ 
?/, ( /i+1) d?„2 
(2n — 1) {2n + 3)/ d/x (2n — 1) (2a + 1) dfx 
This relation will hold good when n = 1, provided we replace (:/P_'^/r//r by zero. 
Thus the right-hand member of (21) is (iqual to 
4aV(l-/r2) ^ 
= 1 
-C„ 
P-l 
+ /0-, 
C 
71 + 0 
(2a, —3) (2a—1) \a,(a-t-1) (2 h,—I) (2a,+3)/ (2/1-1-3) (2?i-|-5) 
The left-hand member may in virtue of (20) be written in the form 
dP 
dfx 
A(i - ^®)2r. 
d/x 
Equating the two members and dividing by 4&>®a’^(l — g^), Ave obtain 
h ^ dP„ 
-- - 
" dix 
a o 
dP„ 
d/x [(2a — 3)(2?/, — 1) 
c. (-f 
2 
a(a-(-l) (2a — 1) (2?/-f 3)/ (2a-f 3) (2?r-f 5) 
+ 
C. 
m+2 
