1914 - 15 .] Spherical Harmonic Computation. 
67 
I represent them shortly in the following way ; 
f{0\, j/45°) = Xv 
1 , 2 , 3, 4, 5 
!/=:(), 1, 2, 3, 4, 5, 6, 7 
The coefficients a, /5 are then expressed by the interpolation formula 
f(0 , 4>) = COS 0 + tt20 COS 2S + ag^ COS 30 + cos 40 
4- (ttjj sin 0 + a^i sin 20 + ag^ sin 30 + sin 40) cos 4> 
+ (jdii sin 0 + fSoi sin 20 + pg^ sin 30 + ^4^ sin 40) sin 
4- (aQ2 + aj2 cos 0 -t- a22 COS 20 + ag2 COS 3$ + COS 40) COS 2<fi 
+ (/3 o2 + ydi2 cos 0 4 /^22 + f^42 
4 - (a^g sin 0 4 - sin 20 + agg sin 30 + a4g sin 40) cos 3</> 
4 (/di3 sin 0 + ^23 sin 20 + sin 30 + sin 40) sin 3<^ 
4- (ttQ4 + tt24 cos 20 + tt44 cos 40) COS 40 
In the first place, all the combinations of the given values of the 
function are to be calculated. They are shown in Table A, being represented 
by the symbols 
[l]i j [^]i 5 [^]i [^^]i 
[1]2J2]2,[3]2 [15^ 
[IhHSJsJSjg [15]g 
This table gives at the same time an appropriate scheme for carrying 
out the calculation. The first column contains the forty values of the 
function arranged in the most convenient order ; the other columns 
explain themselves. It will be seen that numbers which are to be added 
or subtracted stand directly under one another. For these simple opera- 
tions controls are hardly necessary, and are indeed furnished by the mode 
of their summation. The same process repeats itself constantly so as to 
become strongly impressed on the memory. 
The a and 8 follow as sums of products, with constant factors, of the 
numbers just determined. These products are given in Table B. It is 
there evident that for the finding of the thirty coefficients a, (8 (six of 
which are immediately expressible in terms of the others), ninety-two 
products are necessary. Since these can be immediately formed and 
summed by means of the calculating machine, a further control other 
than is furnished in the usual way by the working of the machine is 
superfluous. 
In the second table, for simplification of the numbers the first factors 
with their tenfold totals are set down ; compensation is effected most 
simply by dividing the values of the function by ten before using them, 
whereby as a rule a desirable homogeneity in the whole set of numbers 
is brought about. 
