205] NOTE ON THE SUMMATION OF A CERTAIN FACTORIAL EXPRESSION. 253 
this is the formula in hypergeometric series required for the present purpose, and it 
is certainly true when the series is finite. 
Write now 
« = /3 = r + 1, 7 = r - i ; 
then the first term is [l] r+2 -f- [^]' +1 , which vanishes on account of the numerator, and 
the second term is — \r (r + ^-), and we have consequently 
which gives 
- lr (r + 2)(r + 3) i . bS = - \r (r + |), 
a _ 4r (r + I) 
" (r + 2) (r + 3) ’ 
8 being here the series in r, the sum of which was required, and the particular case 
of Mr Kirkman’s formula is thus verified. It is probable that the general case might 
be treated in an analogous manner by first grouping together the terms which corre 
spond to a given difference x~y, and ultimately summing the sums of these partial 
series; but I have not examined this question. 
2, Stone Buildings, W.G., April 18, 1857.