66 
Proceedings of the Eoyal Society of Edinburgh. [Sess. 
The ani and /3ni are then obtained as sums of the 2^9 (p + 1) products, of 
which the one factor 
1 and the other factor stands in the table. 
Such a table possesses the advantage that a glance enables us to 
recognise and calculate the influence of a change of a given value of the 
function f(^fj.x, upon the coefficients a and ^ ; for if gniw is the value 
in the table for an% which corresponds to X, v, and A/a^ the known change, 
then will gnixvVfw be the corresponding change of anh and 
. P even 
jcOS *<^>1^. 
the change of the function f{0, (p). 
In practical work the direct use of the table is not to be recommended, 
for, although the mode of calculation is indeed very clear, the number of 
products to be formed is great. It is possible also to supply a much 
simpler procedure, since the tabulated values of each are for the greater 
part zero, or equal, or equal and opposite. 
We have now left a few of the different coefficients v^hich are to be 
multiplied by constantly recurring combinations of /^/xx, These latter 
are made up solely out of the sums and differences of the/^/xx, without 
factors, and are quickly formed by calculation with the hand ; all further 
working is best done with the machine. 
In practice the calculator will mostly be concerned with developments 
up to the fourth order of spherical harmonics, and only in exceptional cases 
will be compelled to go as far as the sixth order. I here restrict myself 
therefore to the communication of the formulae and numbers for the case 
p = 4>; their deduction may be left out, as it is quite simple. 
In accordance with the theory, we must take as given values of the 
function the points of section of the meridian 
4> = 0% 45°, 90' 
. . . 315' 
with the parallels whose polar distances are 
^1= 154 58 57-6 
^0= 122 34 46-2 
^3= 90 0 0-0 
57 25 13-8 
2b 1 2-4, 
forty values in all. 
