544 BISHOP TERROT ON THE SUMMATION OF A COMPOUND SERIES, 
Rip p+ 2i) eee pt+qtl Me oe 186" 
Silas ey a ge q-m—q—l...m—q p)+(1x1.2.3...9) 

But the probability of a white at p+q+1" drawing on H, is ao = 

“. probability of white derived from H, is 

ptl.pt2....pt+q+l ini Sve bd 
m+1.m....m—p—qx1.2.3... ig GM Gd) nae) 2. 3 gaa) 
So probability from H, 
re ptl....pt+qtl EN. ti ed. —- 
~ mtil.m....m—p—qx1.2.3. nee q—Smn—q-2;..m—gop—1)x (2.3---¢ +1) 
And so for all the other hypotheses in succession. 
Now this series, omitting for the present the consideration of the fraction 
which is a factor common to them all, is a series of the same form as that summed 
in the last proposition, only that now p+1 must be substituted for p. 
We have therefore the whole probability of a white at p+q+1" drawing 



rie ptl.p+2...ptqrl ps Lead ¥ 
~ m+l.m...m—p—qxl.2...q° pr2...ptqt2 
foi d , ~# 3(H) 

met Lh 21, die dali fone ERY 
Note.—It may be worth observing, that, had we summed the original series in 
Prop. 1. upwards instead of downwards, we should have got for a first factor 
12 LSU 1D A 2 Pie ie) ; 
EAC E OAR EES which must therefore tees Ry OPMENT) And 
that these fractions are equal may be proved independently, for if we divide each 
by 1.2.3...px1.2.3...49, we have on both sides the same quotient 
1 
(ey ee ee aia 
There now remains for solution only the first case of the problem in chances, 
that is, to find the probability of drawing a white ball, when m the number of 
balls is given, and p white and qg black have already been drawn and returned. 
The main object in this case is to sum the series 



Fp se ea) SOON V.m-1" : . : : : eya(1) 
This may be done much as in the preceding case, by taking the successive 
differences of the right-hand factors till the differences vanish, and multiplying the 
successive terms of the last or +1" row of differences into the g+1' summa- 
tion of the successive terms of the series (1+2”. . . +m—1")+(14+2?... +m—2"), 
&e. 

This may be sufficiently explained by going through the operation in a low 
particular case. Let p=2, g=3. 
Se ee ee elle i 
7 i 4 
Rabe NWR cen 
Rabe wei 
