WITH APPLICATION TO SPHERICAL HARMONIC ANALYSIS 
= -- 11;^ cos 2^0 clfjL ; if cr be odd, p odd, n even. 
cos^:)^cZ/x ; if cr „ „ p even, n odd. 
ml = “ [ sin cJ^x ; if cr be even, n odd, n even. 
'lea- -1 
I r+i 
= — R^ sin^;d c//x ; if cr ,, ,, p even, n odd. 
-ff, J -1 
It would tliiis appear to Ije unnecessary to have separate symbols for ul and vl or 
for ml and si, but as it is convenient to tabulate separately the integrals for odd and 
even values ofp, the retention of all four symbols facilitates reference. 
We may now obtain the final coefficients by summation thus : 
l=,l + l 
yO) 
c^vl, 
when 
V. 
is 
even. 
cr 
odd, 
and 
P 
odd, 
i'=i 
i=,- + i 
odd, 
V'-’ 
•C^i, 
5? 
cr 
> J 
P 
oven. 
J, = 0 
i=ii4l 
v(-) 
hlml, 
)) 
n 
J) 
even, 
cr 
even 
> >) 
P 
odd. 
p=l 
) = /i + 1 
y(-) 
?5 
n 
odd. 
cr 
55 
J5 
p 
even. 
p=2 
with similar ecpiations for hi, a and h being ie})laced by a and jB. 
The values of vl, ul, ml and si are given in Tables V.—YIIl. and their logarithms 
in Tables IX.—XII., for values of n, cr and up to 12 inclusive. By means of these 
tables we may, for instance, write down at once the coefficients as far as the third 
degree as follows : 
= 7853986^° 
= •08017567 
g .° = — •21952561'’“b '05857563° 
= - -25974067 + '0493056^° 
c// = ‘0801750; - •340087o/ 
c// = ‘3802290/ - ‘38O22903' 
= ‘I59OGI00' + -3181230/ - -3976530/ 
p7 = -5703456/ - -19011563" 
g3' = -50299067 ~ -2514986/ 
= -0160420/ - ‘410695o 7 + ‘102072o/. 
2 E 
VOL. CC,—A. 
