612 
PROFESSOR SYLVESTER ON THE REAL 
(30) Moreover, we thus see that the average number of variations in an open line 
with ^ positive and v negative signs, which is 
2(2*-l)[>,^-i)+22^F,^ 
will be equal to 
2 %|> ; ^]-2 
22 #([>, v, #-■ |)+|>, v, $ 0 )— 2 [>, v - 9— \) 
;[>> ^]= 
: 22^[jW-, i», ^]=- 
2jU-V 
'|U, + V 
The total number of variations and continuations together is p-\-v — 1. Hence the 
(/*- 
; so that the average 
difference between the two is — (uj 4-v — 1), or — 
P + v vr p + v 
number of variations is greater than, equal to, or less than that of the continuations, 
according as the difference between the numbers of the two sets is less than, equal to, or 
greater than the square root of the entire number of signs. Obviously the average 
should be the same for the variations as for the continuations if the number of signs, 
say n-{- 1, is given, and each is supposed equally likely to be positive or negative. This 
is easily verified ; for multiplying the probable value of each distribution of signs by the 
probable value of the number of variations corresponding thereto, we obtain the series 
I fi / . 1 \ , n/ IV , I\ n , cy, n ^{n + l)n.{n — l) , j n(n+l)2 n ~ 1 n 
(n+l)2»{ 1 • W '( w + 1 )+ 2 (^- 1 )( w + 1 )2 + 3 ( W ~ 2 ) fX3 + * * * j- _ ( ra + l) 2 » = 2 ' 
This is the final average of the number of variations of sign, and will be equal to that 
of the continuations, since the entire number of the two together is n. 
Received October 27, 1864. 
Past III.— ON THE NATURE OF THE ROOTS OF THE GENERAL EQUATION OF THE 
FIFTH DEGREE. 
(31) In a foot-note, Part II. of this memoir, I have shown that when the discriminant 
of the canonizant (constituting an invariant of the twelfth order) of an equation of 
the fifth degree bears a particular sign, the character of the roots becomes completely 
determined by the sign of the discriminant of that equation. 
This has naturally led me to investigate de novo the whole question of the character 
of the roots of an equation of that degree ; and I have succeeded in obtaining under a 
form of striking and unexpected simplicity the invariantive criteria which serve to 
ascertain in all cases the nature of the equation as regards the number of real and 
imaginary roots which it contains ; then passing to the expression for these criteria in 
terms of the roots themselves, I obtain expressions which exhibit the intimate connexion 
between this subject and a former theory of my own relative to the construction of the 
conditions for the existence of a given number and grouping of equal roots, which can 
hardly fail to lead eventually to the extension of the results herein obtained to equations 
of any odd degree whatever. It is the more needful that these results in a question of 
so high moment to the advancement of algebraical science should be made public, inas- 
much as they do not seem to accord with those obtained by my eminent friend M. Hermite, 
who has preceded me in this inquiry in a classic memoir, published in the year 1854 in 
