334 Archdeacon Pratt on a Problem in 



When i is taken indefinitely great, the expression within the 

 brackets ultimately becomes p w , and this is independent of i. 

 For those parts of the body which are outside the sphere (of 

 radius a) u is greater than 1, and the quantity outside the 

 brackets (as I show below) becomes infinite, and therefore the 

 whole expression under the signs of integration is infinite when 

 i is indefinitely increased, unless u=l. As the integral must 

 not become infinite, I come to the conclusion that u = \ or r = a, 

 and the surface of the imaginary body is a sphere and can have 

 no other form. 



2. But I have to show (what I have assumed above) that the 

 quantity outside the brackets becomes infinite when i is infinitely 

 increased. 



P.= coefficient of a* in jl +a 2 — 2ap\ ~% 



or in 



(l-«s)-*(l-^j \ ^=2(^+2) = cos <£, 



in 



( I+ i„ + ^^ + ...X. + i; + fe|5 + ...) 



= 2A, cos i <£ + 2 A j_ 2 cos (i— 2)<£ + ... 

 Put i(f> = yjr, then 



P;=2A.cosi/r + 2A i _ 2 cosM — -W -f ... 



or in 



1.3..(2t- l)/^ , 1\ L 1.3..(2t-3)1 

 2.4..2i 



Now 



A,- is less than 1 and greater than — .: 



1 & 2i 



U 



2t-2 " 2' 



A, (if i is odd) 1 „ .— • j— $ 



A (if? is even) 1 



11 



i i 



Choose yjr less than 90°, and increase i indefinitely ; then cf> is 

 indefinitely small, or p nearly 1, or the attracted point c,\x ] ,w l 

 (which may be anywhere outside the body) is always taken im- 

 mediately above the point r, fi, a> on the surface. All the terms 

 of the series for P* are positive, and therefore P* is greater than 



