1914-15.] Spherical Harmonic Computation. 
65 
and combine the known coefficients in the C and S with the % when we 
obtain 
A=p + 1 v = 2p-l 
X = 1 v = 0 
A=p+1 v = 2p-l 
B»<= 2 Z 
A = 1 1^ = 0 
/ i7T\ 
•/( 
7T \ 
P*’ v; 
/ ^7^\ 
•/( 
^ 7T \ 
( 2 ) 
In every case the 2p(p + l) coefficients and (of the latter p + 1 
are ah initio equal to zero) are tabulated for each combination n, i, and the 
operation to be carried out with the calculating machine is then continued 
quite simply so that each of the given values of the function 
is multiplied 
with the corresponding G and H respectively, and 
the sums of all products taken. 
After determination of the and the interpolation formula for 
the function f{juL, <p) becomes 
= (PooAqo + PiqAio + .... + PpoA^o) 
+ (PiAn + P21A21 4 - .... + P^,iApi) COS </) 
+ (kiibn + k-2ib2i + .... +P^jtBpPsin^ 
+ (b22^22+ .... +~Pp2Ap2) C0s2(p > 
+ (P22B22+ .... +P^2Bij 2) siu 2^ 
+ 
^ cos 
(3) 
where the associated spherical harmonics P are functions of the powers 
of sin^ and cos^( = ya). For convenience of application, the expressions 
within the brackets in (3) require to be changed into rows which are 
developed in sines and cosines of multiples of 0 ; that is, the arrangement 
takes the form 
/ (0, 4>) = (ttoo + 0-10 cos ^ 4- Ooo cos 2^ + . . . . 4- Opo cos pO) 
4- (oii sin 0 4- ogi sin 2^4- + 0.^1 sin pO) cos </> 
4- (fSii sin 0 + /321 sin 2^4- + Ppi sin j;(9) sin cf> 
4" (oq 2 4“ 0^2 cos 0 + 0.22 COS 2^4".... 4" 0.^,2 cos pO^ cos 2(f> 
+ (/?02 + /di 2 cos 0 4- ^22 cos 2^4-.... + (ip 2 cos p6) sin 2(^ 
+ 
The second step to be made in preparing once for all for the carrying 
out of the calculations is that, instead of the above-named tables for the 
G„^ and H„i, similar tables may be immediately constructed for the calcula- 
tion of the a^i and This is easily possible, since the a^i, /3ni are simple 
known functions of A„^ and 
VOL. XXXV. 
5 
