175 



In same way the probability derived from H^ is the same frac- 

 tion into 



(m —q—1 . m — q — 2 m — q—p — 1) x2.3 S'+l 



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

 we have the whole probability equal — 



(m+ 1 . m"-m—p — q) xl .2.«.g p -\-2 . p + ^"'p-\- q-\-2 



-, P + 1 



m+1 .m m — p — q = 



^ ^ p-{-q-h2 



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

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

 ball at the p + q + V^^ trial, depends entirely upon p and q, and is 

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



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

 and the balls drawn are replaced. 



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

 series 



m-l'^x V + m-2\P . 2^ P . m-l'^ 



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

 and the sum found to be 



2,m-ll^ + c^2 2^m-2'^ d^^^m-q^P 



Where l^m—V^ means the q^^ integration of the series 



1 + 2^ m+1'^ 



and c?j, d^f 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 with the a + j8 + y, &c., of the last case, 

 the probability of a white at the p-\- q + V^^ drawing is 



^2+im-lip^i + d^ S^ ^tm- 2li>+i d^ S. + im-gl^+i 



m (2^m-ll^ + c^2 2,m-2l^..7:7:^^^,^rr^ 



If m be infinite, this becomes — i+L_^ _ JP_Ji 



mS^+^mp p + g + 2 



The following Gentlemen were duly elected as Ordinary 

 Fellows : — 



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



2. The Rev. John Gumming, D.D. 



