94, 95] NEWTONIAN POTENTIAL FUNCTION. 183 



Multiply both sides by cos m (ay <) dco and integrate from 

 to 2?r. 



/27T oo r-Zn 



( r 3) F(&>)cosra(ft> $)d = 2 T n 



Jo o Jo 



cos m o> 



Every term on the right vanishes except the rath which is equal to 

 7rT m ((/>). Accordingly we find for the circular harmonic T m the 

 definite integral 



(14) T m ($) = i 



For m = 0, we must divide by 2. 



Writing for T m (</>) its value 



A m cos m<f> -f 5 m sin ra$, 



expanding the cosine in the integral, and writing the two terms 

 separately, we obtain the coefficients 



1 /* 2jr 1 f 27r 



(15) J. = ^ I F(ft>)c?a>, A m = V(a>)cosmct)dco, 



^TTJo 7TJO 



This form for the coefficients was given by Fourier*, who assuming 

 that the development was possible, was able to determine the 

 coefficients. The question of proving that the development thus 

 found actually represents the function, and the determination of 

 the conditions that the development shall be possible, formed one 

 of the most important mathematical questions of this century, 

 which was first satisfactorily treated by Dirichlet"!'. For the full 

 and rigid treatment of this important subject, the student should 

 consult Riemann, Partielle Differentialgleichungen; Picard, Traite 

 d' Analyse, torn. 1, chap. ix.J 



95. Spherical Harmonics. A Spherical Harmonic of degree 

 n is defined as a homogeneous harmonic function of the coordinates 

 x, y y z of a point in space, that is as a solution of the simultaneous 

 equations 



* Fourier, Theorie analytique de la Chaleur, Chap, ix., 1822. 



t Dirichlet, " Sur la Convergence des Series Trigonometriques, " Grelle's Journal, 

 Bd. 4, 1829. 



I A resum6 of the literature is given by Sachse, Bulletin des Sciences Mathe- 

 matiques, 1880. 



