AND IMAGINARY ROOTS OF EQUATIONS. 
613 
the ninth volume of the Cambridge and Dublin Mathematical Journal, since which time 
I am not aware that the subject has been resumed by any other writer. The discrepancy 
between our conclusions may be only apparent ; but there can be no doubt of the supe- 
riority of the form in which they are herein presented, inasmuch as only three functions 
of the coefficients are required by my method, and five by M. Hermite’s. The solution 
offered by M. Hermite is confessedly incomplete, but to this great analyst none the less 
will always belong the honour, not only of having initiated the inquiry, but of having 
emitted the fundamental conceptions through which it would seem best to admit of suc- 
cessful treatment. The arrow from my hand may have been the first to hit the mark, 
but it was his hand which had previously shaped, bent, and strung the bow. 
Our methods of procedure, however, are widely dissimilar, and by employing my well- 
known canonical form for odd-degreed binary quantics, long since given to the world, I 
have succeeded in evading all necessity for the colossal labours of computation required 
in M. Hermite’s method, and am able to impart to my conclusions the clearness and 
certainty of any elementary proposition in geometry, not scrupling to avail myself for 
such purpose of that copious and inexhaustible well-spring of notions of continuity which 
is contained in our conception of space, and which renders it so valuable an auxiliary to 
Mathematic, whose sole proper business seems to me to be the development of the three 
germinal ideas — of which continuity is one and order and number the other two*. 
Section I. — Preparation of the General Binary Quantie of the Fifth Degree. 
(32) Let (a, b , c, d, e , ifx, ff— F(#, y\ 
a cubic covariant of F is the canonizant C, where C represents the determinant 
a b c d 
bed e 
c d e i 
y 3 —y 2 x yx 2 —x 3 . 
Let us first suppose that this form does not vanish identically, and has at least two 
distinct factors |, r\ linear functions of x, y, where of course f, yj are each of them 
determinate to a constant factor pres; giving any value to the constant factor for either 
of them, we may write F(x, ^)=<E>(g, ??)=(«, (3, y, S, e, ?i) s , and the canonizant of <P 
with respect to |, tj becomes the determinant T, where T represents 
a (3 y $ 
(3 y & s 
y § s 
v 3 -r. 
* Herein. I think one clearly discerns the internal grounds of the coincidence or parallelism, which observa- 
tion has long made familiar, between the mathematical and musical eOos. May not Music be described as the 
Mathematic of sense, Mathematic as Music of the reason? the soul of each the same ! Thus the musician feels 
Mathematic, the mathematician thinks Music, — Music the dream, Mathematic the working life — each to receive 
its consummation from the other when the human intelligence, elevated to its perfect type, shall shine forth 
glorified in some future Mozart - D irichlet or Beethoven-Gauss — a union already not indistinctly foreshadowed 
in the genius and labours of a Helmholtz ! 
4 n 2 
