140 a, 141] SPHERICAL HARMONICS. 393 



This form for the coefficients was given by Fourier 1 ), 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. 2 ) For the full and rigid treatment of this 

 important subject, the student should consult Kronecker, Theorie der 

 einfachen und der vielfachen Integrate; Picard, Traite d 'Analyse, Tom. I, 

 Chap. IX; Riemann-Weber, Partielle Differentialgleichungen ; Poincare, 

 Theorie du Potentiel Newtonien. 9 ) 



141. Spherical Harmonics. A Spherical Harmonic of degree n 

 is defined as a homogeneous harmonic function of the coordinates 

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

 equations 



84) 



Q*\ i , 



85) x -a + V -o-H- ^ -Q = n V. 



dx v dy dz 



The general homogeneous function of degree n 



a nQ x n + a n ^ lt0 x n - 1 y + a n - 2 ,<>x n ~ 2 y 2 h 



-f an-Litf*-^ + a n - 2 ,ix n - 2 y^ f- 



contains 1 + 2 + 3 + n + 1 = terms> ^ gum of itg 



second derivatives is a homogeneous function of degree n 2 and 

 accordingly contains 2 ' n terms. If the function is to vanish 



identically, these 2 l coefficients must all vanish, so that there 



(n 1) n , .. n (n 4- 1) (n 4- 2) . , 



are - ~ relations among the - ^ - coefficients of a harmonic 



of the n ih degree, leaving 2^ + 1 arbitrary coefficients. The general 

 harmonic of degree n can accordingly be expressed as a linear func- 

 tion of 2n + 1 independent harmonics. 



1) Fourier, Theorie analytique de la Chaleur, Chap. IX, 1822. 



2) Dirichlet, "Sur la Convergence des Series Trigonometriques 1 ', Crelle's 

 Journal, Bd. 4, 1829. 



3) A resume of the literature is given by Sachse, Bulletin des Sciences 

 MatMmatiques , 1880. 



