AND IMAGINARY ROOTS OF EQUATIONS. 
585 
second, and third, as well as the third, fourth, and fifth coefficients form geometrical 
series ; hence it is obvious that the transformed equation may be reduced to one or the 
other of the two following forms, viz. 
a A + iex 2 y + Qe 2 x 2 y 2 — iexy 2 + < y 4 = 0, ( a ) 
or 
0 , ( 6 ) 
with the condition in the latter case that e 4 — e 2 is positive, i. e. e 2 >l. 
be one real value of m, and when D is negative, a real value of im. Tbe former case possesses over tbe latter a 
striking distinction, which is that all the roots of m will he real ; for, as I have shown elsewhere, if m is one root 
1 — 2m 1 -\-2m . 
the complete system of roots will he +m, + p %~ n , + Y—3 m : m ^ a ^ er case the reahty of the two values 
+ im does not seem necessarily to imply the reality of the other 4 valued of the system. 
( c ) Analogy suggests the establishment of an analogous canonical form or forms for ternary cuhics, of which, 
as is well known and is even dimly foreshadowed in Newton’s Enumeration of Lines of the Third Order, the 
theory runs closely parallel to that of binary quartics. This will be effected by assuming the form 
F(x, y, z) = + 3eLoc 2 y + Qgxyz, 
and assuming g so as to make the discriminants of 
dF dF dF 
dx dy dz 
all zero. This gives rise to a quadratic equation in g, of which the roots areg=e, g= 2e 2 — e. When g=e, I find 
S=e(l— e) 3 , T = (1 — e) 4 (l + 4e — 8e 2 ), A=T 2 +64S 3 =(l+8e)(l-e) 8 . 
When g=2e i —e, I find A = (1 — e)*'(l — 4e)i(l +2e) k , where i,j, h are integers to he determined. These forms 
will, I think, be found important in the future perspective discussion of curves of the third degree. Whilst I 
yield to no one in admiration of the surpassing genius with which Newton has handled these curves, I cannot 
withhold the expression of my opinion that every theory of forms in which invariants are ignored must labour 
under an inherent imperfection, and that Newton, from want of acquaintance with the indelible characters which 
their invariants stamp upon curves, has in the parallel which he has drawn between the generation by shadows 
of all conics from a common type, and of all cubic curves from a limited number of forms, either himself fallen 
into error of conception, or at least used language which could scarcely fail to lead others into such error. Eor 
no species whatever of cubic curve can be formed for which an infinite number of individuals cannot be found 
which defy linear or perspective transformation into each other ; whereas all conics proper may he propagated 
as shadows from a single individual. It should be noticed in connexion with this subject, that the indelible 
s 3 
characters of quartic binary, and cubic ternary forms are two in number, viz. the value of - (where s, t are the 
two fundamental invariants in either case) and the sign of t. The indelibility of the sign of s being implied in 
s 3 
the invariability of the value of - , does not constitute a distinct character. Of course all symmetrical invariants 
have an invariable sign ; but this is not the case with skew invariants, as ex. gr. M. Heemite’s octodecimal inva- 
riant of a binary quintic, which will change its sign with that of the determinant of transformation. 
( a ) Whilst upon this subject of invariants, I may allow myself to make a remark bearing upon what will be 
noticed further on in the text about a ease of equality between roots not necessarily being a mark of transition 
from real to imaginary roots. If a, b, c, d being the roots of a binary quartic we form a secondary cubic, of 
which the roots are (a— b)(c— d), (a—c)(d—b), (a—d)(b—c), it may be easily shown that two of these quan- 
tities become equal, or, in other words, the roots of the original equation mark out a harmonic group of points 
when t (the cubinvariant) is zero. Notwithstanding which a change of sign in t will not command a change of 
character in the above three roots of the secondary (nor consequently of tjie original equation), because it is not 
an odd but an even power of t, viz. f, which enters into the discriminant of the secondary. 
