173 



Monday, 21st February 1853. 

 Sir T. M. BRISBANE, Bart., President, in the Chair. 

 The following Communication was read :— 



On the Summation of a Compound Series, and its applica- 

 tion to a Problem in Probabilities. By the Right Rev. 

 Bishop Terrot. 

 The series proposed for summation is 



+ „^^ZT.T;r:^7^2 m-^p X2.3.4...5 + 1 



+ p.p-l.p-^ 1 x«^.i^»^=^~l-"^-P+i+l 



In which series each line or term is the product of two factorials, the 

 first consisting of p, the last of q factors of successive numbers. And 

 in each successive term the factors of the first factorial are dimi- 

 nished each by unity, and the factors of the last increased. 



The method employed to sum this series is to multiply the sum 

 of all the left-hand factors into the first right-hand factor ; the sum 

 of all except the first, into the difference between the first and se- 

 cond of the right-hand factors, and so on ; thus reducing the series 

 to the form 



-X_x O^TT^l . ^^q m-^Ti+ 1) X 1 . 2 . 3.. .5-1 



p + l 



If this integration on the one side and differentiation on the other 

 be continued for q times, the series is reduced to the single term 



_gj^^ ^^ '<t~^ ^ xm + l.m m-7+i> + l- 



p + l .p+2 p + q+l 



This summation is applicable to the solution of the problem, Sup- 

 pose an experiment concerning whose inherent probabdity of success 

 we know nothing, has been made fTq times, and has succeeded 



