606 
PROFESSOR SYLVESTER ON THE REAL 
unknown linear functions of x : thus we are entitled to say that in general the number 
of real roots in such an equation is not the maximum four, but a less number. This 
remark is of importance, as showing that on this subject it is possible to speak with 
scientific certainty, and on other than empirical grounds, of what may in general be 
expected to take place. Thus we find Newton declaring twice over in the chapter 
quoted, that in general his rule will give not merely the maximum, but the actual 
number of the imaginary roots in an equation. I am strongly inclined to doubt the 
truth of this assertion; but it is important to be satisfied by analogy that such an 
assertion may rest on a scientific and demonstrative basis, and not on the utterly falla- 
cious foundation of arithmetical empiricism ( 2S ). 
( 28 ) A few additional words on this question of probability may not be unacceptable. In order to meet the case 
of the degree of the superlinear form or equation being odd as well as even, let it be supposed known under the form 
the values of the quantities Cj being supposed to be left wholly indeterminate, and only the signs of the quanti- 
ties X to be given. Let w be the inferior number of like signs in the X series, meaning thereby that the num- 
ber of signs of one sort is to, and of the other sort u>, or more than to. 
Let the probability oT the specific maximum of real roots being 21c when m is even, be represented by y> 2 *, 
and of its being 27c + 1 when m is odd by 7r 2 / c +i ; also let s 2 *, o- 2 / f+1 represent the number of cases when w and n 
are given which correspond to the specific maximum being 2k, 2Jc +1 respectively. Suppose u=l, then obvi- 
ously, when m is even, we have s 2 =n, j? 2 =l. But when n is odd a t =2 (for when either extreme element alone 
is negative the trans-rotatory cycle has the variation-index unity), and <r 3 =n—2, so that 
2 n-2 
Again, suppose w—2, m being even; then obviously s 2 is the number of contiguous duads in a cycle of n 
elements, and s i is the remaining number of duads ; hence 
so that 
n — 1 n—3 
iv 
2 n - 3 
—V 2*—n— 1* 
2nd. Suppose w=2, m being odd, so that <r v <r 3 , a. will have to be separately estimated. To fix the ideas, 
let the X series be termed a, b, c, d, e, f, g, in which two of the elements are supposed of one sign, say negative, 
and the rest of the opposite sign, say positive ; then the only dispositions of sign which correspond to the specific 
maximum being 1 are those in which a, b or else/, g are both negative. Hence u x —2. Again, the dispositions 
of sign which make the specific maximum equal to 3 are those in which a, g are both negative, those in which 
a and c, d, e, or f are negative, those in which g and e, d, c, or b are negative, and, finally, those in which 
any two contiguous elements except the a and g are negative. Hence <r 3 =l + 2(rc— 3) + («— 3)=3n— 8 ; and 
it should be observed that this result cannot be prejudiced in its generality by the supposition of any of the 
components of a 3 becoming negative, since w=2 implies that n is at least 4. Hence, finally, 
so that 
4 6)1—20 n~ — 7n + 16 
1 n 1 — n 71-3 tt 2 — n ’ n 2 — n 
This example serves to show how much more difficult is the computation of the respective probabilities when m 
is odd than when rn is even, owing to the break of continuity in the cycle of readings on passing from the last 
to the first term. 
