210 
PROFESSOR A. SCHUSTER OX SOME DEFINITE INTEGRALS, 
The calculation of the ten coefficients g up to the third degree involves therefore the 
determination of sixteen Fourier coefficients. There is no h coefficient of zero type, 
and there are therefore only six h coefficients up to the third degree which depend 
upon twelve Fourier coefficients. In general there will he \ (n +1) + 2) 
g coefficients, and |-(w + l) n, h coefficients up to and including the degree, ghdug 
a total of {n + 1)’^ coefficients in the series of sjoherical harmonics. To determine 
these completely, it would he necessary to calculate {n +1)” Fourier coefficients 
for the quantities g and n n 1 for the quantities /q giving a total of (n + 1) (n +2) 
Fourier coefficients. The tables accompanying this paper being calculated for 
values of n up to 12 are therefore sufficient to determine 169 coefficients in the 
series of s})herical harmonics, by means of a simple computation after the deter¬ 
mination of 182 coefficients of the Fourier series. These latter coefficients may be 
evaluated by mechanical devices. 
§ 12. Special Application to the Theory of Terrestrial iMagnetism. 
The advantages of the proposed method of obtaining the coefficients of a series 
of spherical harmonics are considerably increased when the quantities to be repre¬ 
sented l)y such a series are not given directly, but by means of their differential 
coefficients. This is the case when the magnetic potential has to be calculated by 
means of the observed magnetic forces. If X and Y represent the comjDonents of 
magnetic force resolved towards the geographical north and east resjiectlvely, the 
magnetic potential is determined l)y means of the ecpiations 
clY/dd = X ; dVIdcj, = - Y sin 9. 
If no electric currents of sufficient intensity traverse the earth’s surface, a 
function Y can be found wliich satisfies both equations. If Y sin 6 be obtained in a 
series proceeding l)y spherical harmonics, then all the terms which depend on the 
longitude are at once expressed in a siinilar series, as the integration according 
to (f) leaves each term in the standard form. The treatment of the other component 
acting along the meridian involves, however, serious difficulties, and it is not 
necessary here to enter into the question as to the more oi' less complicated methods 
by means of which V has been hitherto derived in the standard form from X. 
The method based on the results of the preceding investigation avoids these 
difficulties. For odd values of cr, X is expressed in a series, each term of which 
has the form cos n6 sin w'i<^ or cos nO cos while for even values of cr, X is 
expressed in terms of the form sin n6 sin and sin nd cos m^. After integration 
with respect to 6 , the formula of § 11 will determine the required coefficients. Y e 
may treat tlie eastern force similarly, obtaining a series proceeding by cos pd or 
sin p6 according as cr is odd or even. Y sin 6 is then derived in a series proceeding 
