542 BISHOP TERROT ON THE SUMMATION OF A COMPOUND SERIES, 
Integrating then each line separately, we have the sum 
qd Et 
p+] SHgtl.m—g. . ie Me— Paget Leds (Bievs, ate ane q-1 
+ 9g —— 
pti ™—-g-m—q-1 Ee ee m—p+qx2.3.4....... qd 2@ 
+ q° ee 
p+ m—q—1l.m—q—2...m—p+q—-1x3.4.5...... q+1 
&e. &e. 
If again we treat this form as we have done the original, by taking the 
differences of the second factorials as they now stand, and again integrating, we 
reproduce the sum in the form 
.q—1 ph es. \ 
pil pee M442. m—q+1 -.-+.m—p+q+1x1.2.3....9q—2 
D) 
.q-1 ¢ 
+ eT pee Mat m—q > o miGeo bob m—pt+qx2.3.4....q—1 
&e. &e. 
It appears, then, that we may continue this differentiation on the one side 
q times, and integration on the other g+1 times; and that at each succeeding 
operation, an additional next lower factor will be introduced into the numerator 
of the fractional coefficient, and an additional next highest into the denominator. 
And after q¢ differentiations, the last factorials will all become unity; and, the 
middle factorial having acquired an additional higher factor at each of g+1 inte- 
grations, we have for the sum of the series— 
-g—l.q—2....1 — 
pal pit apres em 5 oN m—q+tpt+1 } Rabies yl) 

I. 
The Problem in Probabilites to which the foregoing summation is applicable, 
is the following :— 
Suppose an experiment concerning whose inherent probability of success we 
know nothing, has been made p+gq times, and has succeeded p times, and failed 
q times, what is the probability of success on the p+q+1'™ trial. 
This Problem is interesting, because it tends to the discovery of a rational 
measure for those expectations of success which constitute the motive for a large 
portion of human actions. The force of such expectations commonly depends, 
not upon reason, but upon temperament; and, according, asa man is naturally 
sanguine or the reverse, so in all the contingencies of life, does he over-estimate 
or under-estimate the chances in his favour. 
It would be going much too far to think, that we can give an algebraic formula, 
by the application of which a man may, in every practical case, correct his 
natural tendency to error, and arrive at a strictly rational amount of expectation. 
a 
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