602 
PROFESSOR SYLVESTER ON THE REAL 
completely express such term, conceive every ray affected with a distinct sign + or — . 
A pencil thus drawn with its rays so polarized will give a complete representation of 
any given superlinear function, and may be called its type-pencil ( 23 ). 
I am now able to state the following proposition : 
(18) The number of real roots in a superlinear equation cannot exceed the variation - 
index of its type pencil , regarded as a per-rotatory type , if the degree of the equation be 
even, and as a trans-rotatory type if the degree of the equation be odd. I prove this 
inductively as follows. 
1. Suppose the theorem to be true when the variation-index of the type-pencil is 
not greater than the even number v, and consider an equation of the odd degree (2«'-|-l), 
for which the type-pencil viewed as trans-rotatory has the variation-index p+1. 
Let a phase of this type be taken, say corresponding to the rays § n , § n _ 1 ... &, such 
that the per-rotatory type obtained by striking out the term has the variation-index v 
(as we know may be done by virtue of the Lemma). 
Take for new axes 0£', Of when 0|' coincides with § x ; then it is clear that the 
pencil g n _ 1 . . . g 2 , will still serve as a type-pencil to the given function, the only 
change being that some of the rays, namely those that did lie on one side of f 15 have 
been inverted in direction and changed in sign (corresponding to a change in the coeffi- 
cient a, b, accompanied with a change in the sign of the corresponding g), whilst the 
rays on the other side of g x have been left unaltered. 
The points (a„ b x ), {a 2 , b 2 ) . . . ( a n , b n ) corresponding to the rays g x , g> 2 , . . . g n will, with 
respect to the new axes, change their values, becoming converted into (a 15 0), (ce 2 , /3 2 ), 
(a 3 , /3 3 ), . . . (a„, j3 n ), where (3 2 , /3 3 , . . . (3 n will still all be positive, the angle between § x 
and g n being the same as between the two extreme rays in the original figure of the type- 
pencil, and the superlinear equation may now be written in the form 
F(w, v)=i l {u l uf +l -\-z 2 {ot 2 u-\- /3 2 v) 2i+i -\-s 3 (oi 3 u-^-(3 3 v) 2i+1 -]rs n (a n u-\-l3 n v) 2i+1 =0, 
where u, v are real linear functions of x , y. 
( 23 ) Let a circle be imagined pierced by a pencil containing any number of rays protracted in both directions, 
say in the opposite points a, a ; b, ; c, y ; d, 8 ; and let these points, taken in order of natural succession from 
left to right, or right to left, be a, b, c, d, a, /3, y, 8. Then, commencing with any point c, a complete circulation 
will be represented by the succession of transits 
do d, d to a, a to /3, /3 to y, y to 8, £ to a, a to b, b to c. 
But whether a, /3, y, 8 bear respectively the same signs or signs contrary to those of a, b, c, d, the transit be- 
tween any two points /3 to y will be of the same nature, as regards continuance or change of sign, as the transit 
from b to c, and thus we see that the complete cycle or total revolution above indicated is only a reduplication 
of, and may be fully designated by the hemicyclic succession c to d, d to a, a. to /3, /3 to y, for which the num- 
ber of variations therefore will be the same as for any similar succession obtained by commencing with any other 
element in the original system of points instead of c. If the opposite points bear like signs, the above succession 
of transits may be indicated by the order c, d,a,b,c; if they bear contrary signs by the order c, d, a, b, c, and thus 
it is that the idea arises of the two kinds of so-called circulation, but which are in fact only more or less dis- 
guised species of semicirculation. 
