390 
MR. W. D. NIVEN OR CERTAIN DEFINITE INTEGRALS 
2i / f ^-1) 3, v 
0,U\ n\\^ OYOo _ 1 “H • • •) 
i\ ir r 2(2i — l) 
1( K 1 r* 
~ (p+jv cos xpydxp, - 
7T J n 77" J 
J 0 (ji +jv cos y) m 
^0 1 ) 2 2 
Now by definition 
/ g—*2 2, I 
P“ 4 P ^ + 
.. . . \ .r {+1 d l - a V <P , d c 
(b -)=(-i)‘7rnrh+ 
(B) 
(C) 
P) 
t! dz \dg drjJ\/z^ + ^7) 
By Theorem ii. this is 
1 , d'- 17 / d°-Q. 
r-— 2 COS o-fbv^-r V 1 cr ——’ 
2n! r dz \ dp? 
We observe that, in this result, the axis of the harmonic Q; is the line OP, as explained 
in the proof of Theorem i. We must therefore change the axes of coordinates, 
making the new axis of 2 coincide with OP. 
■_ <Z°P; 
If r l w h en expanded, becomes 
Ay--_. . . 
w r e have to consider the effect of operating upon this with 
There will result 
that is 
d . d V— 
cos a— — sin a — 
dz dp] 
i — cr\ (A cos 7 <r a — B cos 7 a 3 a sin 3 a -f- . . .) 
<PP; 
l — cr ! 
dp? 
where /x is now the cosine of the angle between OP and the axis of z. Hence we 
obtain the first and second forms of the harmonic corresponding to the forms (A) and (B) 
of the zonal harmonic, viz. :— 
(^ cr) — WT7 2 cos aov 
v ’ ' 2 a i\ r dp? 
2 it . , . 
— 0 -+ ., 2 COS cr<i>W u} 
2(2»-l) 
(а) 
( б ) 
Let us now consider the definite integral 
