610 
PROFESSOR SYLVESTER ON THE REAL 
The integral must satisfy the further condition that [//<, 1, g] shall be unity when g is 
1, and zero for all values of g greater than 1. 
Assume the value of [//», 1, g] obtained by the method given in art. (21). This 
obviously satisfies the initial conditions corresponding to g=l. Moreover we may easily 
deduce from it the equalities 
(g-Vg 
(v-l)v 
O, »-l, y-l]= ( -- _^ + i )(y _ ff) 0, *-l, 9l and 0, r, ?] = fct+> _ I)( ,_ g) [>, ?]• 
Hence the equation in differences will be satisfied if it be true that 
(v-l)v 
v-g 
(—1 +9)+ 
(g~ l )g 
? 
v-g 
which is obviously the case, since v 2 — v— g 1 — g—{v— g)(v-\-g— 1). 
Since, then, the assumed value of v, g~\ is correctly determined when v—1, it is 
obvious, from the form of the equation, that it holds good] for all other values of v, as 
was to be shown. 
(26) From the equation 
0, v, + i] _ y—g){v—g) 
[p> g] ~ g(g+ 1) 
making {^—g){ v ~g)—g{g J r^-) or (!— ^j rV j r p we ma y readily infer that the value of g 
for which the probability \jjj, v, g\ is greatest is the integer part of p if that quan- 
tity is non-integer, or the quantity itself and the number next below it (indifferently) if 
it is an integer. 
(27) If we apply a similar method to [po, v, g-\~i], we obtain by aid of the formula 
above given, 
+ 2f4v— (^+y)y (^ + 1)- v(y + l— y) ■ 
[?>’', g — i] 2y,v + p + v- (it + v)y 7* 
and equating this ratio to unity, we obtain 
2 g.V — (fJ, + y)ry y 2 . 
2/ttv+ft+y— (p+v)y («.+ l)(v + 1) — (f* + v + 2)7 
or writing gj-j-v=_p, [Av=q, 
(f +y>)y 2 - ( 3pg-+ 4 q+f +p)y+2 q(q+p + 1) = 0. 
The roots of this equation will be both of them real, for its determinant is 
ft + 1 + 1 6q 2 +(p 2 +_p 3 )(^ 2 + J' 2 ), 
which is necessarily positive. Hence it follows that there are two positive roots of the 
equation. Whether there will exist values of g which give actual maxima or minima 
values, or one and the other to [gu, v, g-\~Y\, depends on the further condition being 
satisfied that the values of g in the above equation shall come out, one or both of them, 
not greater than either of the two numbers v. The inquiry connected with the satis- 
faction of this condition may be conducted by means of repeated applications of the 
