GAUSS'S THEOKY OF TERRESTKIAL MAGNETISM. 113 



this depends on - or {(x a) 2 +(y &) 2 + (z c) 2 }"*. For 



x, y, z y the ordinates of the experimental magnet, put 

 their values (already used) r . sin u . cos X, r sin u . sin X, 

 r cos u. And for a, b, c, the coordinates of a disturb- 

 ing particle of magnetism, put similar coordinates, 

 a = r Q . sin U Q . cos X , b r Q . sin U Q . sin X , c = r . cos w . 

 (If the experimental magnet be on the earth's surface, 

 and the disturbing magnetism be within the earth, r Q 



is always less than r.) The value of - now becomes 



[r 2 2r r {sin u. sin u . cos (X X ) + cos u . cos u ] + rf]~* ; 

 which can be expanded in a converging series 



where T Q = 1, and T v T z &c., are functions only of u, u , 

 and X X . Put R for the earth's radius (the symbol r 

 being still reserved for the radius at the place of ob- 

 servation, in order to preserve the generality which 

 admits of differentiation with respect to r). Then V 



or 



ISp . - may be put in the form 



where tfP 9 = - f T . $p, 

 f T^ r 2 Sfi, &c. The general term will be 

 B n+ *P n 



r n+l > 



where RP n = -/T n .r n . 8/*. 



