420 TREiMBLEY. 



This result as we have said Trembley obtains, though he goes 

 through more steps to reach it. 



Suppose however that before effecting the integration with 

 respect to x we use the following theorem 



1 4 X , ^'(^'-1) orJ^ q'{q'-V){q'-^) x^ , 



/ + 1 1/ + 2'^1.2 y + 3 1.2.3 _p' + 4 



= (^ ~ ''^' liTTT+T ^ (/ + 2+1) (/ + 2') i^^ 



(P' + 2' + 1) (/ + 2 ) (/ + 2' - 1) (1 - •'^) 



^_ 2'(2'-l )(2'-2) ^ , ' 



(/ + 2+1) {P + 2') (/ + 2' - 1) (P +2-2) (l-.^)^ . 



Then by integrating with respect to x, we obtain 

 \qj\-q | j9+p' + l f 1 9^' ^+y + g + g +^ 



y)4j/+£+</+2 Ip'+^'+l (/^'■+^'+l)(/+^') ^ + 2 



q{q~l) (7^+7y+g+g'+2)(^+/+g+g+l) 



It is in fact the identity of these two results of the final inte- 

 gration which Trembley assumes from observing its truth when 

 q = 1, or 2, or 3. 



With regard to the theorem we have given above we may 

 remark that it may be obtained by examining the coefficient of a?*" 

 on the two sides ; the identity of these coefficients may be estab- 

 lished as an example of the theory of partial fractions. 



774. Trembley then proceeds to an approximate summation 

 of the series ; his method is most laborious, and it would not repay 

 the trouble of verification. He says at the end, Series haec, quae 

 similis est seriei quam refert Cel. la Place ... He gives no refer- 

 ence, but he i^robably has in view the Hist, de VAcad Paris 



for 1778, page 310. 



775. We have next to consider a memoir entitled Recherches 

 SUV une question relative au calcid des prohahilites. This memoir is 

 published in the volume for 1794 and l79o of the Memoir es de 



