200 
MR. ARTHUR SCHUSTER ON THE 
Such products as p i -~ and p may give rise to terms independent of X only 
dk elk 
through products of the form cos X cos (X — a), cos X cos (X + a) or cos X sin (X — a), 
cos X sin (X + a). 
We need therefore only consider that part of p which involves cos X and terms of 
the first type in xfj and S. Selecting the two terms {k'„ cos (X — a) +p/ 2 cos (X + a)} Q' n 
in the expression for S and rejecting all terms not containing X, we are left with 
cZS 
_pd S_ 
sin 6 cIX 
= \y {Pn — K'n) Q'n sill a. 
Similarly 
e ( J± 
de 
p cos v -~j-~ — -ism 
6 cos 6 sin a. 
do 
By means of the formuhe of transformation previously given we find 
sin 6 cos 6 
dQ'm _ m.(m+ 1) 2 , m 2 + m— 3 
. V / .. TTW, on — 
_______ Q/ m 2 (m+ 1) 0/ 
m—2 ' / ci iWo / o m + 2* 
dd (2m+l)(2m—1) " (2m—l)(2m + 3) (2m+l)(2m+3) 
In the special case worked out in the previous pages in which m — 1, 
■ioQ'i-isQ 
-l sin 6 cos 6 = -WO'.-AO' 
dU 
The velocity potential \fj 2 2 contributes, as far as the term p cos 0 is concerned, 
nothing to the zonal harmonics in R. 
We have therefore in the second case, and whenever the velocity potential does not 
contain a term of the first type, 
^ = («'.-/.) Q'.sm «. 
If R, as far as its zonal harmonics are concerned, is written in the form 2p„°P„ sin a, 
we find, as — —= sin 6 = Q' 
d0 dp 
Pn = h (x'n-p' n ) .(44). 
We should have obtained the same value if we had applied the general equation to 
this case. When r = 1, v/e have to consider the terms depending on c ~, and must 
therefore write 
For all other values of n equation (44) applies. 
