AND IMAGINARY ROOTS OF EQUATIONS. 
587 
Observation . — To make the foregoing demonstration quite exact, it should be noticed 
that when the criteria L, M, N have been brought to the form 1-0, and the series 
of substitutions of y-\-z x for y has set in, we have 
N=0, SN=0, m=(cd-be)e, W=Ns=0, S 8 M=0. 
Consequently if cd — be should become zero, we can no longer go on decreasing M. But 
as soon as cd—be— 0, since we have also d 2 —ce, b , c, d, e come to be in geometrical pro- 
gression, and the transformed equation takes the form 
ax * + 4 ux\j + 6u 2 x 2 y 2 + 4 afxy 3 + u*x i = 0, 
with the condition u 2 —au 2 negative, or a>l. Hence we have q t x i +(x-\-ay) i = 0, which 
obviously has all its roots impossible ( 13 ). 
(6) We may now pass on to equations of the fifth degree, in which the case resisting 
induction will be that where the criterion-series bears the signs 
- + + -• 
Let the criteria be called L, M, N, P, so that writing the equation 
ax 5 -J- 5bx 4 y-\-I 0ca?y 2 + 1 0dx 2 y 3 + 5 exy 4 -\-fif = 0, 
L=5 2 -«c, M =c 2 -bd, N =d 2 -ce, Y=e 2 -df, 
and writing for x, x-\-zy, we have, as before, 
SL=0, m=(bc-ad)e, M=Ls 2 , 
so that M may be continually diminished. 
If M becomes zero before either N or P changes its sign, the criterion-series for the 
transformed equation becomes — 0-1 , and for its derivative in respect to x, the series 
is 0 + — , which proves the existence of four imaginary roots in the transformed, and 
consequently also in the given equation. In like manner, if N becomes zero before M 
or P have changed their signs, the criterion-series becomes — -f- 0 — , which obviously 
leads to the same result. So likewise the same inference may be drawn if L and M, or 
M and N, or L, M, N become zeros all at the same time, and we have only to consider 
the case when, L and M retaining their signs, N becomes zero. At this moment the order 
of the substitutions must be reversed, and for y must be written y-\-zx ; we shall then have 
P=0, SP=0, m={de-cf)z ; 
( 13 ) From the first and third criteria it follows that in the form (a, b, c, d, ej£x, yy, a, c, e have the same sign 
. b 2 d 2 
and may be regarded as all positive ; so that writing a — —=h 2 , e — — =P, the form becomes h 2 x 2 +F-\-Jc 2 y 2 , 
where 
F = ■ — x* + 4 ba?y + bca?y 2 + 4 dxy 3 + — y 4 , 
and consequently the given form will have all its roots imaginary when this is true for F, so that we might 
have proceeded at once to deal with the forms marked (a), (b) at p. 585 ; but as the method of homographic 
transformation by infinitesimal substitutions appears to be necessary in passing to the corresponding forms 
in the case of the fifth degree, and as in treating that case reference is made to what appears above, I have 
thought that no object would be gained by altering the text. 
MDCCCLXIV. 4 K 
