AND IMAGINARY ROOTS OF EQUATIONS. 
611 
processes of Sturm’s theorem ; but I shall not enter upon it, as it appears to lead to 
calculations of complexity disproportionate to the interest of the result. 
(28) It may be noticed that the average value of [fA, v, g~\ can be calculated without 
any difficulty. This will be 2(</[fA, v, < 7 ]), or 
7r(,x,+ 
7r/x.7rv 
(f,- i)(v_r 
1 
7r(ju. + v — 2) 
■7TV ["n , (fA— l)(v — 1) , (ft — l)(ft — 2 ) (v — l)(v 2) , -] 
v-l)L + 1 + 1 . 2 2 +**-J 
ftV 
7r(ft + v— 1) ?r(ft — 1 )tt(v — 1) (/a + v — 1)’ 
so that the average number of variations of sign in a per-rotatory pencil with fA positive 
and v negative signs is — or a little more than the harmonic mean between fA, v. 
In like manner, for a trans-rotatory pencil this number will be 
2(2?+l)|>, >, g+ i]=D», », fl + 2((2j+l)(^-l)[> > ,, y]), 
which, observing that 2 [ft, v, g] = 1, and (ft, v, + 2 ^ - ^ =2, gives as the average 
number of variations of sign pi. 
& 1A + V ft + v-1 
fA + V ft + 1 
Received May 10, 1864. 
(29) The simplest mode of calculating the value of [ft, v, g ] is the following: 
Let [ft, v , <7), |jt, v, (g — -|-)j denote the probability that an arrangement in open line (in 
which, as is the case in applying Des Cartes’s rule of signs, no account is taken of the rela- 
tion of the extreme signs to each other) shall contain respectively 2 g and 2 g — 1 variations. 
Conceive a circular arrangement of y groups of positive and y groups of negative signs. If 
this circle be opened out into a line at an interval between a positive and negative sign (of 
which there are 2y), one variation will be lost; but if at any of the remaining fA-J- v — y 
intervals, the number of variations remains unaltered. Hence we derive immediately 
l>> 9)= 
fA + V- 
p+ 
rW>y] and 
But we may find [ja, v, g — by counting the arrangements which give fA, v, 2 g — 1 
variations of sign. These may be all obtained, and without repetition, by intercalating 
every distribution of fA into g groups with every distribution of v into the same ; and the 
intercalation may be performed in two ways, according as the parcels of the fA signs, or 
those of the v signs, are taken first in order. Hence we have 
r 1 \_ 2 (f 4 ~ 1 )(^- — 2 ) (fA— g + l) (»—!)(» — 2) ... (v— ff + 1) ■ny.-rcv 
2) ~ 1.2 ... (ff-1) 1.2 ... (g-1) 7r(^ + v) 
27TjU.7T (fA — 1 ) 7TV7T (v — 1 ) 
— 7r{(^ + y)w(g-l)r{g— !)■*([*— g)*{y— g)’ 
and thus 
r -l ' r I-! '■ f* « X)-nrn\* J- j 
\-t /j,t>, 9j 2 g 9 2 J 7r(m + v+ l)7ign[g — l)7r (jut. — g)ir(y — g) ’ 
7 — l)7rv7r(v — 1) 
as previously found ; also 
r v \ (^ + v-2^VfA7r(fA-l)7 r v7r(v-l) _ 
4 N 
MDCCCLXIV. 
