584 
PROFESS OK SYLVESTER ON THE REAL 
in the course of this reduction, either N retains its sign or changes it ; and if the latter 
is the case, N must have passed through zero. If when M becomes zero N is still nega- 
tive, the criteria of the linearly transformed equation become — 0— ; and it may be 
noticed that its first, middle, and last coefficients must have the same sign, by virtue 
of the negativity of the two last criteria, and the second and fourth the same signs, by 
virtue of the zero middle criterion ; consequently the equation will take the form 
(X 2 + e 4 )x 4 + 4<? 3 s x 3 y + beh*ot?y* + 4 e?xy 3 +(+ 2 + g 4 )^ 4 = 0 , 
or 
XV+jtA 2 ^ 4 +(^4;gy) 4 =0, 
which obviously has all its roots impossible. This being true of the transformed equa- 
tion, will also, on the suppositions made, be equally so of the original equation. 
Let us next suppose that N changes its sign either at the instant when, or before M 
becomes zero. If M and N both become zero together, so that the criteria of the 
dl? 
transformed equation bear the signs — 0 0, calling the transformed equation F=0, 
will have all its roots equal, and F will therefore be of the form (ax-\-by) 4 -\-lcx 4 , with 
the condition (a 3 bf-(a 4 -\-/c)(a 2 b 2 )< 0. 
Hence Jc is positive, and consequently F=0 has all its roots imaginary ; and the same, 
as before, must hold good of the original equation /==0. 
It remains then only to consider the case when N becomes zero before M vanishes. 
When this is the case, as soon as N is reduced to zero, in lieu of the substitution of 
x-\-zy for x, we must leave x unaltered, and continue substituting y-\-zx for y. We 
thus start from the sequence 0 ; N will then always remain zero, and we must 
either come to the series — 0 0, which we know, from what has been shown above, cor- 
responds to four imaginary roots, or to the sequence 0 + 0, which I shall proceed to 
consider. 
Since the first and last coefficients must have the same sign, we may, by giving 
either variable a proper multiple ( u ), make these two coefficients alike, and with the first, 
(ii) (a) The f orm ("1, e> e) y~y m ay be regarded as a new and, for many purposes, useful canonical 
form of a binary quartic. It may be made to comprise within its sphere of representation all forms correspond- 
ing to two or four imaginary factors, but excludes the case of four real factors. The ordinary canonical form 
(l, 0, 6m, 0, 1]$+, yY comprises within its spheres of representation those forms for which the factors are all 
real or all imaginary, but, so far as real transformations are concerned, excludes the case of two real and two 
imaginary factors [that case is met by the form 1, 0, 6m, 0, — 1 '£oc, ?/) 4 ], as may easily be established either 
by decomposing the form first named into its factors, or by the consideration that its discriminant A is 
(1— 9m 2 ) 2 , and is therefore always positive; whereas if a form which it is used to represent have two real 
and two unreal factors, its discriminant is negative. If now the determinant of transformation be D, and the 
discriminant corresponding thereto be called A', we have A'=D 6 A, showing that D 2 is negative, and the trans- 
formation therefore unreal. 
( b ) The reality of m for each of these cases (usually assumed without proof) may be demonstrated as follows : 
Calling the cubic invariant and the discriminant of any cubic form T, D, we shall have, using the ordinary canonical 
form, (~ j V _ 9 m 2 ) 2 =ip showing that when D is positive, which is the case of four real or unreal factors, there will 
