AND IMAGINARY ROOTS OF EQUATIONS. 
583 
If all the criteria are zero, it is evident that, whatever n may be, all the roots are real. 
In every other case we shall find that zero may be made positive or negative at will. 
Thus in the case before us, if the two criteria are 0+ or 0 — , there will be a pair of 
imaginary roots, as the first may be read as (- and the second as . 
To prove this, we have only to observe that in either case ^ will have two equal roots; 
so that /will be of the form {ax-\-byf -\-cy 3 , which obviously, for any real values of 
a, b, c , has only one real root. 
(5) We may now pass to the case of n= 4, and excluding for the moment the con- 
sideration of zeros, limit our attention to the criterion series 1 . 
Let ax 4 + ibx 3 y -f- 6 cx^y 1 + idxy 3 -J- e/= 0 be the equation for which the signs of the 
criteria b 2 —ac, c 2 — bd, d 2 —ce are — 1 — . Call these criteria L, M, N respectively. It 
has to be proved that all four roots are imaginary, since there are two distinct negative 
sequences, each sequence consisting of a single — . Let x become x-\-zy{^°), where g is 
an infinitesimal quantity, and transformed into one between u and y ; then we have 
obviously, 
e$«=0, hb=ae, hc=2bs, }>d=3ce, le=4:ds, 
&L= 2 bhb — dbc = 0, &M = 2 che—bM — dlb —{be — ad) g, 
m=(Mc+c^-«M)g=2(5 2 -«c)g 2 =2Lg 2 ; 
so that ei 2 M is essentially negative, since L is so. 
Hence, by continually augmenting x by an infinitesimal variation, we may, leaving L 
unaltered, so choose the sign of g as to decrease M : nor can this process stop when be— ad 
becomes zero, by reason that £ 2 M is negative. Hence we may reduce M to zero. Now, 
Also we have 
A = 1 + 4(s 3 + ij 3 ) — 6ey — 3e 2 ij 2 
>1 + 4(e + — 6sij — 3 e 2 ij 2 
>1 — 6sij + 8(£ij)f — 3f 2 ij 2 ; 
or, writing sr)=q 2 , A > 1 — 6q 2 + 8q 3 — 3 q*, 
>(l-q)\l+3q); 
but 1>9'>0. Hence A is positive. 
Hence in either case two of the roots of the cubic are impossible. Or the same thing may be shown more 
immediately from the identities 
a 2 A -(a 2 d+2b 3 - 3 abef + 4 (ac - b 2 f, 
d 2 A = ( ad 2 + 2c 3 - 3bcdf +4 (bd-c 2 ) 3 , 
so that A must he positive, and therefore two roots imaginary, if either bd,x? or ca>b 2 . It may he noticed 
that the square and cube in these identities are semi-invariants, being in the first of them unaffected by the 
change of x into x-\-7iy, and in the second by the change of y into y + hx. 
(i°) qq^g me thod of infinitesimal substitution is that which I applied in my memoir “ On the Theory of Forms,” 
in the Cambridge and Dublin Mathematical Journal, to obtain the partial differential equations to every possible 
species of invariants (including covariants and contravariants) of forms, or systems of forms, with a single set or 
various sets of variables, proceeding upon the pregnant principle that every finite linear substitution may be 
regarded as the result of an indefinite number of simple and separate infinitesimal variations impressed upon 
the variables. M. Aronhold has erroneously ascribed to others the priority of the publication of these equations. 
