AND IMAGINARY ROOTS OF EQUATIONS. 
605 
ing to the j group of s are separated by one or more of the rays with a contrary sign to 
themselves. Thus it appears that when only the units , s 2 , . . . s„ are given, we may 
impose a maximum upon the number of real roots in the superlinear equation ; this 
limit may be called the absolute maximum , being the double of the inferior number 
of like signs in the series s lf e 2 , . . . s n when the degree is even, and one more than such 
double when the degree is odd ( 26 ). 
The specific maximum , on the other hand, will depend on the form of the type-pencil, 
and cannot be ascertained until the coefficients of the linear elements are given. It can 
never exceed, but may be less than the absolute maximum. It may, indeed, be easily 
proved that in general the specific maximum will be less than the absolute maximum. 
Thus, by way of example, suppose the degree to be even, and the inferior number of 
like signs to be 2 ; the absolute maximum number of real roots will be four, but the 
specific maximum will more generally be only two. For let the number of linear terms 
in the superlinear function be 2-} -n, n being 2 or any greater number; and first, to fix 
the ideas, suppose n= 2. The type-pencil, which is to be read per-rotatorily, consists of 
four rays, say a , b , c, d, following each other in uninterrupted circular order, of which 
two are to bear positive and two negative signs. If the two negative signs fall on a, c 
or on b , d, the variation-index will be 4, but in the other four cases of incidence such 
index will be only 2. Consequently the chance is 2 to 1 ( 2T ) that the specific maximum, 
which may be 4, is not greater than 2 ; and consequently the chance that there will be 
four real roots in the equation will be only a chance (too difficult to be calculated, but 
which is a function of the degree of the equation) of the chance that there will be as 
many as four real roots in the equation u n l -\-ul—ul—u n i = 0, where u 1 , u 2 , u 3 , u 4 are 
( 26 ) (a) jf a SU perlinear form of an odd degree contains an odd number of terms, say 2k 1, the greatest value 
of the inferior number of like signs is Tc, and the extreme limit to the number of real roots mil be 2&+1. 
If it contain an even number of terms, say 21c, the greatest value of the inferior index is lc ; but for this par- 
ticular case it mil readily be seen that a limit may be assigned to the variation-index closer than that given by 
the rule in the text; in fact the variation-index cannot in that case exceed 21c — 1 , which will therefore be the 
extreme limit to the number of real roots. Now suppose the canonizant of an odd-degreed function of x, y to 
have all its roots real, then it may be expressed by a superlinear form of which the number of terms will be 
2i+l or 2i, according as the degree is 4i+l or 4i — 1. In the one case the number of real roots cannot exceed 
2i+l, in the other 2i — 1. Hence the following somewhat curious theorem : 
( b ) If the canonizant of an odd-degreed quantic in x, y, of the degree 4i + 1, has no imaginary roots, the quan- 
tise itself must have at least i pairs of imaginary roots. From the fact that when the roots of the canonizant of a 
quintic are all real there must be one pair at least of imaginary roots, we can infer that when the discriminant 
of a quintic is positive and that of its canonizant is negative, the equation has one real and four imaginary roots. 
This observation has led to a long train of reflections, which will be found embodied in the 3rd part of the 
memoir. 
( 27 ) This, in fact, is identical in substance with the noted problem of determining the chance that two straight 
’ines drawn on a black board will cross. Hr. Cayley, of whom it may be so truly said, whether the matter 
he takes in hand be great or small, “nihil tetigit quod non ornavit,” suggests the following independent proof 
of this. Taking unity as the length of the contour, fixing the extremity of one of the lines, and calling s the 
distance of its other end from it measured on the contour, the chance of the second line crossing this is easily 
seen to be 2s(l — s), which, integrated between s=0, s=l, gives -I, as before obtained. 
4 M 2 
