304 Prof. J. J. Sylvester on Spherical Harmonics. 



Since every spherical harmonic of integral degree is a linear 

 function of the differential derivatives of (# 8 +y 2 + 3 8 )~*, the 



whole theory of the diplo-spherical-harmonic-surfaee integral 

 is contained in the annexed equation, which springs immedi- 

 ately from the expression found above for the bipotential of a 

 spherical surface at two internal points (slightly modified by 

 taking — h, —k, —I; —h f , — //, — V for the coordinates of 

 the points) by means of the simple and familiar principle that 

 any differential derivative with respect to x } y, z of a function of 

 a, y, z is identical with what the corresponding derivative with 

 respect to h, k, I of the like function of x + h, y-\-k, z + l be- 

 comes when h, k, I are made to vanish. 



Let U stand for ic i -2Vih / . u 2 + 2A 2 . 27i' 2 , and let 



V(A, ,, I ; V , M, !>£•( *, i £)*$ *, J) *,, 



where C I> and ^ are forms of function which denote series, 

 whether finite or infinite, containing only positive integer 



powers of the variables. Then, ifp = — -= and cl$ is 



r ' */x 2 +y 2 + z 2 



the element of a spherical surface of unit radius, the complete 



integral 



= 4ttV(0, 0, 0; 0,0,0). 



When <£> and SP are homogeneous forms of function, each 

 of the degree i, if we write 



T = l-22M' + 2/* 2 .2// 2 , 

 and make 



a ( n, k, i ; v, v, *>* ( * , *, |) * (A, fa * ) --L , 



the value of the corresponding harmonic-surface integral be- 

 comes 



2^n(o, a, o,o, o,o). 



I am not aware that a rule for finding such integral so 

 simple in form and of such absolute generality in operation as 

 the one above has been given before ; the interesting rule fur- 

 nished by Professor Clerk-Maxwell, i Electricity and Magne- 

 tism ' (vol. i. p. 170), assumes that.<X> and '\f r have been each 

 reduced to the form of the product of linear functions of 



—, -j-} - a reduction which cannot practically be effected, 



(' su a*y ccz 



