Mr. W. D. Niven. [Dec. 11, 



, denote 



the expressions comprised under 



G = (1, x, y, z, yz, zx, xy, xyz) Q l ---- 0*, 



where any of the quantities inside the brackets is the multiplier of 

 the product of the 9's outside, will satisfy Laplace's equation, provided 

 n equations of the form 



P g r | 4 4 = 



are satisfied, where p, q, r are respectively 3 or 1 according as G does 

 or does not contain x, y, z as factors. 



2. If K r denote _ +_J'l_+_! , so that K r = Q r +l, then, in 



a 2 + 0, 6 2 -r0 r c* + 0, 



like manner, the expressions comprised under 



H = (1, a% y, z, yz, zx, xy, xyz) K^ ---- K, 



will satisfy Laplace's equation for precisely the same values of 6 as in 

 1, and it may be shown that there are 2 + l independent conjugate 

 H-harmonics of any degree n. 



3. The function H is a spherical harmonic. Suppose it is of the 

 nth degree and of order a, and let it be denoted by H,,'. The corre- 

 sponding ellipsoidal harmonic, i.e., for the same values of 0, may be 

 denoted by G,*, and it may be shown that G n * and H,,' are con- 

 nected by the relation 



D 2r 1 



4.C_iy _ " __ L IH* 



' 2->!(2n-l)(2n-3)....(2n-2r+ir J 



where D* = a 2 ~ + b* ^-+c 3 ?L 



ox oy Oz 



4. Let iryc be any point on the surface of the ellipsoid and x'y'z' 

 the corresponding point on a concentric sphere of unit radius, so 

 that 



x = ax', y = by', z = r', 



then will Q r (x, y, 2) = -0 r K r (', y', *'), 



and 



Q(x,y,z) = (1, a, 6, . . . . , a6c) (-^) (-0 2 ). ... H (',!/', 2')- 

 By means of thase relations any function/ (x, y, z) or/ (ax 1 , by', cz') 



