TREMBLEY. 419 



for 1778, page 304, for 17S3, page 419; and Theorie . . . des Proh. 

 page 381. 



Trembley says, 



Supponit Cel. la Place nates esse Parisiis intra certnm tempus, ^) 

 puA-os q puellas, Londini autem intra aliud temporis spatiiim p' pueros 

 q puellas, et quaerit Probabilitatem, causam quae Parisiis producit 

 pueros esse efficaciorem quam Londini. E supra dictis sequitur hanc 

 Probabilitatem rejDraesentari per formulam 



X' 



{l-xyx'P'([-xy'dxdx' 



y (1 - xf x'' (1 - x'Y dxdx' 



Trembley tben gives the limits of the integrations ; in the 

 numerator for x from a^' = to ic' = x, and then for x from a? = 

 to x = l\ in the denominator both integrations are between 

 and 1. 



Trembley considers the numerator. He expands x'^ (1 — x'Y in 

 powers of x and integrates from a?' = to x = x. Then he expands 

 x^ {1 — xY and integrates from a; = to a? = 1 ; he obtains a result 

 which he transforms into another more convenient shape, which 

 he might have obtained at once and saved a page if he had not 

 expanded x^ (1 — xY- Then he uses an algebraical theorem in 

 order to effect another transformation ; this theorem he does not 

 demonstrate generally, but infers it from examining the first three 

 cases of it ; see his page 113. 



We will demonstrate his final result, by another method. We 

 have 



jx [L x)ax-x ^^'^i lp' + 2^ 1.2 p' + S J 



Multiply by x^ (1 - xY and integrate from x = to a; = 1 ; 

 thus we obtain by the aid of known formula 



[q \p+p +1 ( 1 ^'1 p +y + ^ 

 p+p'+q + ^ I^TTi ~ r y + 2 ]r+/T7T3 



q' jq' - 1) 1 (P + P+^)(P + P+^) 



"^ 1.2 y + 3 (p+2^' + 2 + 3)(P+/ + 2 + "^) 



27—2 



