175 

 In same way the probability derived from H, is the same frac 



tion into 



(^ -q-\ . m-q-2 m-q-p-l) X 2 . 3 ?+ 1 



and so for all the other hypotheses. And summing again this senes, 

 we have the whole probability equal— 



p+1 . P + 2—P + 9 + 1 ^ ^^_1_^- — -J^ ^ 



m+1 .m 'n-p-?-p + 5 + 2 



As this expression does not involve .n, it follows that when the 

 balls drawn are not replaced, the probability of drawing a white 

 ball at the p + q+ 1'*^ ti-ial. depends entirely upon p and ?, and is 

 unaffected by the magnitude of m, whether finite or infinite. 



The last portion of the paper considers the case where m .s given, 

 and the balls drawn are replaced. 



It is evident that in this case the main point must be to sum the 



series . 



^;^Zal^xP + m-2l''.2^ P.m-1" 



This was effected by a process similar to that used in the last case, 



and the sum found to be 



2^-;ir:n:i'' + <i2 2,^»rr2"' d^.^m-q^" 



Where 2,.ir=T"' means the q^'^ integration of the series 



1 + 2" m + 1'", 



and d„ d,, d,, &c., mean the 1st, 2d, 3d terms of the q^^ row of 

 differences of the series 1, . 2% &c. „/...!.„ 



Applying this as was done withrtie« + ^ + 7. &«•. ^^ '^' ^^ '^''' 

 the probability of a white at the p + ? + 1'^ drawing is 



~~;r(^7^in:i'' + d2 2,m-2l'' d,j\^,m-q'^) 



If m be infinite, this becomes ^"^TTT^ -f+ ^ + 2 



The following Gentlemen were duly elected as Ordinary 

 Fellows : — 



1. James M. IIog, Esq. of Newliston. 



2. The Rev. John Ccmmino, D.D. 



