AND IMAGINARY ROOTS OF EQUATIONS. 609 
When <7=0 the above expression fails ; but reverting to the equation from which it is 
derived, we obtain 
(23) These combined results admit of an easy corroboration, for 
Hence the equation marked * gives 
7r(^ + v)^V + V 
+ 
W ^ l>* v > ffl -i ; n , V 
7rfjL7rv 7r(ju. + v) 
g)TtgTt{y-g)%g~ 
which is true, since the left-hand side of the equation is ^ + ••• ? 
which is obviously the coefficient of x v in + l ) v , i. e. in (l+^) ,i+ ‘'. 
(24) If we wish to find the chance of the specific superior limit becoming equal to the 
absolute superior limit, we must write g in the above formulae equal to v, that one of the 
two quantities (a, v which is not greater than the other, and we shall obtain 
r -i — 1) 
b V J 7r(/X-|-V — l)7r(|X — v)’ 
r I in 7r[X7r([x, — l) 
[^’ I '+a]--(-- +v ^ ( ”~ v ZT) 
so that, in fact, \ja, v, v-\-^]=\ja, v+1, t'+l], which relation may also be obtained by 
a priori considerations. 
(25) With reference to the remark made concerning the mode of obtaining the value 
of \ja, v, g\ I proceed to show how it may be obtained directly by the integration of an 
equation in differences, and by a method analogous in idea to that by which \ja, v, 
was made to depend on \ja, v, g~\. For as in that case we conceived an open pencil to 
be closed and then reopened, so we may imagine one of the rays to be withdrawn and 
then reinserted. In this way, observing that the effect of introducing a negative sign 
into a circle of (a positive and n negative signs consisting of v distinct groups of each is 
to produce no change in the number of the groups if inserted between two negative 
signs, but to increase that number by unity if inserted between two positive signs, we 
may infer that the probability of v becoming v-j-1, in consequence of such insertion, is 
and of v remaining unaltered, is 
p + v b ’ g.+n 
Hence we obtain the equation in differences, 
l>> b g]= 
v— 1 +g 
fA + V —1 
Db #] + 
p-g+i 
p + v—1 
l>> ^ — !]» 
in which g, may be considered constant, and v and g to vary. 
