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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 Eev. 

 Bishop Terrot. 



The series proposed for summation is 



m— g. m — q—\...m — q-\-jp-\-\ xl . 2 . 3...g 



+ m — q — l . m — g — 2 m — q^p x 2 . 3 . 4:...q-\-l 



+ p . p—1 . p — 2. 1 X m—p . m—p-\- l...m— p + ^+ 1 



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 diff'erence between the first and se- 

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

 to the form 



~_ , X (m — q+1 . m — q m—p + q+1) x 1 . 2 . 3. . .g— 1 



H ^-^i'^^ — q ,m — q—l m — p + q) x 2 . 3 q 



&c. &c, &c. 



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 



g,g—l,q — 2 1 - _ 



— ^-zr^ ^ r xm + 1 .m m—^q + p + l. 



p + l.p-\-2 p + q+l 'd-rr-T- 



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

 pose an experiment concerning whose inherent probability of success 

 we know nothing, has been made p + q times, and has succeeded 



