262 Archdeacon Pratt on a Problem in 



the external attraction of which is zero. It is required to 

 find the form and law of density of this imaginary body. Let 

 r, 6 (cos #=//,), co be coordinates to any point of it, p the density 

 of that point, c the distance of the external attracted point from 

 the origin of r, p the cosine of the angle between r and c, r the 

 radius to the surface of the imaginary body, and a the mean 

 value of r. Then P< being Laplace's coefficient of the ith order, 



fTf : ^M± =0 by hypothesis; 



Jo J -J. vW^-aoy ' ™ 



{ r 2 ^3 . r t+2 "I 



^P + Jp,+ ...^P«+ ... JMndr^O. 



ra:< 







This is true for all values of c, 



/»2tt n\ r*r 

 Jo J-iJo 



pr^VidcodfMdr^O. . . . (1) 



p is a function of r, <o, fi; but as its value must be independent 

 of any particular unit of measure for r, f must occur in it with 

 another symbol, as r or a, as a divisor. Let r + r— v; then p 

 may be taken to be a function of v, co, p; 



.\ j T pr ( +*dr= -r^f pv i+2 dv 



1 fv i+3 ^ 



~r*+ 3 U + 3 P (i + 3) 



dp 



(i + 3)(»+4)«fo 



v i+5 d*p \ 



+ (i + 3) (i + 4) (i + 5) dv* m :J [V ~ l)i 



by successive integration by parts 



- J_ f^L _ P n) + p (2) fcc 1 , 2 > 



-ji+a \ t - + 3 ( t - + 3)( t - + 4 ) t (j + 3)(i + 4)(,- + 5) °""./'W 



where jo (0) , /o (1) , /o (2:) , ... are values of p and its differential coeffi- 

 cients with respect to v when v = l, and are therefore functions 

 of co and ft only. If the differential coefficients of p do not va- 

 nish after a certain number of differentiations, their numerical 

 coefficients will become so small as to make them insignificant. 

 Suppose that p expanded in a series of Laplace's functions is 



p=R + R 1 .p 1 +R 2 .p 2 + ... +R n .p n + . . . , 



where R , "R lf R 2 , ... are independent of co and jjl, and are func- 

 tions of r only. 



When the set of operations indicated within the brackets in 

 (2) is performed on R , let the result be called <j> -, and so let 



