relation to the Figure of the Earth. 263 



X , 2 , . . . <f> n represent the series which result from the same 

 set of operations performed on R x , R 2 , . . . R n . Then 



! 



1 



P rt+2dr =Zi+r {<f>o+<l>i-Pi+ ...+*n.f>«+ • ••} 



In this, r = a x some function of o> and p, is the equation of the 

 sought surface. The condition (1) shows that when this last 

 formula is multiplied by Pi and integrated, the result must be 

 zero ; which amounts to this, that when the formula is expanded 

 into a series of Laplace's functions, it must have no ith function : 

 let it therefore equal 



^+r(M +(i-l)M 1 + (i-2)M 2 + ... +(i~w)M n + ...), 



which is such a series in its most general shape ; M , M D ... M„ 

 may themselves be functions of i (if necessary), but not in such a 

 way as to cancel the coefficients written above, viz. i— -1, i— 2, . . . 

 i—n. Hence, by equating these last two series, slightly trans- 

 forming, and then extracting the (z' + 3)rd root, 



r _ / 0p + 01 -/>i+<fr 2 .p 2 + •'• + 0»'P» + >•> li+3 



a LM +(i-l)M l + (t-2)M a + ...+(i-n)M n + ..J 



0o ri ; hZ 0o 



i — 1,, i — 2^ T i — n 



__M J + - M -M 2+ ... + — 



1+ V : M J + V : M 2 +...+ ^M,+.., 



-^> ^, . . . are all functions of i ; and so are the coefficients of 



0o 0o 



Mj, M 2 , . . . whether i enters into M , Mj, M 2 , ... or not. 



Hence when the fraction containing these two series of Laplace's 

 functions in the numerator and denominator is expanded, and 

 then its (i + 3)rd root expanded, and the whole arranged in a 

 series of Laplace's functions, every term will have some function 

 of i involved in it ; and the series for r-Ha, which should represent 

 the equation to the surface of the imaginary body, will be a func- 

 tion of i. It ought, however, to be altogether independent of i 3 as 

 is very clear. Hence the arbitrary quantities must be so chosen 

 that i shall disappear. The only way of doing this is, first, to 

 make p v p 2 , . . . p n and Mj, M 2 , . . . M n all zero; this makes p 

 independent of ca and pu, and a function of r only ; secondly, to 

 make O =M O , or r=a; that is, the bounding surface of the 

 imaginary body is a sphere. Hence the only solution which 

 satisfies the problem is, that to preserve the external attraction 

 the same the materials of a body can be rearranged solely by 



