
( 541 ) 
XXXIV.— Summation of a Compound Series, and its Application to a Problem in 
Probabilities. By BisHor Trrrort. 
(Read 21st February 1853.) 
The series proposed for solution in the following paper is— 

(m—q.m—q-1l..... m—gqt+p+1)x(1.2.3...... q) 
+(m—q—1.m—q-2... m—qtp) x(2.8.4 .....q+1) 
: oe OY 
si ((Dal Jae l ato popota bs Oks 1 x(m—p-m—p+1..m—p+q+)) 

The law of this series is manifest. Each term is the product of two factorials, 
the first consisting of p, and the latter of g factors. And in each successive term, 
the factors of the first factorial are each diminished by one, and those of the latter 
increased by one. 
Let there be a series, X,Y,+X,,Y,+ ........... X, Y, 
where, Y,=Y,+4,,Y,=Y,+4,=Y,+4,+A,, and so on. 
Then the series=X, x Y, 
4+X,_,xY, +4, 
+X,» x Y¥,+4,+4,, 
&c. 
=2XK,x ¥,+2X,_.x 4,+2X,_, x a, + &e. 
where 3 X, means the sum of all the terms of X from X, to X, inclusive. 
Let us then, in the first place, take the differences of the second factorials— 
= eS ESCROW eee ee qt)=(2.3.4...... +4 
—(2.3.4....q+1)4+(8.4.5...... Qi) (Sibi De ie q+1).¢ 
&e. &e. 
Hence the sum of the whole series= 
= (m—q.m—q—-1....... m—p+q+1).1.2.3:.... q—l.qg | 
+3(m—q-1.m—q—2....... M—pHig) 2. Sick. ee qq (B) 
+23 (m—q—2.m—q—3..... pres deer oaks Dae ey” WANs WRI a qtl.g | 
&c. &c. 
VOL. XX. PART IV. 7G 
