210 SIMPSON. 



uppermost in a single throw is equal to the ratio which the spheri- 

 cal surface bounded by the moving radius bears to the whole 

 surface of the sphere. Thus the problem is reduced to finding the 

 area of a certain portion of the surface of a sphere. 



868. Simpson gives two examples of the Summation of Series 

 on his pages 70 — 73, which he claims as new in method. 



(1) Let {a + xy be denoted hj A-\-Bx-\- Cx^ + Dx"" + . . . ; 

 required the sum of 



A Bx Cx' 



1.2...r"^2.3... (r + l)"^S.4...(r + 2)"^**'* 



' Integrate both sides of the identity, and determine the con- 

 stant so that both sides may vanish when a? = ; thus 



{a + x Y^' g"-^^ _ . Bx^ Cx^ Bx"^ 

 71 + 1 n-\-\ 2 3 4 



Repeat the operation ; thus 



(?i + l)(n+2) n + 1 (n+l)(7^ + 2) 



_A^ B^ C'^ Dx^ 

 ~'1.2"^2.3"^3T4"^4.5"^"" 



Proceed thus for r operations, then divide both sides by a?*", and 

 the required sum is obtained. 



(2) Required the sum of 1" + 2« + 8" + . . . + ic". 



Simpson's method is the same as had been already used by 

 Nicolas Bernoulli, who ascribed it to his uncle John ; see Art. 207. 



869. Simpson's Problem xxix. is as follows : 



A and B, whose Chances for winning any assigned Game are in 

 the proportion of a to 6, agree to play until 7i stakes are won and 

 lost, on Condition that A, at the Beginning of every Game shall set 



the Sum p to the Sum ^x-, so that tliey may play without Disad- 



ct 



vantage on either Side; it is required to find the present Value of all 

 the Winnings that may be betwixt them when the Play is ended. 



The investigation presents no difficulty. 



