[A2TO.:3 
XXXIII. Note on Mr. Jerrard’s Researches on the Equation of 
the Fifth Order. By A. Cayuny, Esq.* 
» | elena of the same set of quantities which are, by 
any substitution whatever, simultaneously altered or si- 
multaneously unaltered, may be called homotypical. Thus all 
symmetric functions of the same set of quantities are homo- 
typical: (v7+y—z—w)? and wy+zw are homotypical, &e. 
It is one of the most beautiful of Lagrange’s discoveries 
the theory of equations, that, given the value of any function of 
the roots, the value of any homotypical function may be rationally 
determined +; in other words, that any homotypical function what- 
ever is a rational function of the coefficients of the equation and 
of the given function of the roots. 
The researches of Mr. Jerrard are contained in his work, ‘‘ An 
Essay on the Resolution of Equations,’ London, Taylor and 
Francis, 1859. The solution of an equation of the fifth order is 
made to depend on an equation of the sixth orderin W; and he 
conceives that he has shown that one of the roots of this equa- 
tion is a rational function of another root: “The equation for 
W will therefore belong to a class of equations of the sixth 
degree, the resolution of which can, as Abel has shown, be 
effected by means of equations of the second and third degrees ; 
whence I infer the possibility of solving any proposed equation 
of the fifth degree by a finite combination of radicals and rational 
functions.” 
The above property of rational expressibility, if true for W, 
will be true for any function homotypicalwith W ; and conversely. 
I proceed. to inquire into the form of the function W. 
The function W is derived from the function P, which denotes 
any one of the quantities p,, po, p3. And if x, Xo, Ly, L4, Lz are 
the roots of the given equation of the fifth order, andif a, 8, y, 6, € 
represent in an undetermined or arbitrary order of succession 
the five indices 1, 2, 3, 4, 5, and if e denote an imaginary fifth 
root of unity (I conform myself to Mr. Jerrard’s notation), then 
Pv Pa Ps; and the other auxiliary quantities ¢, u, are obtained from 
the system of equations— 
* Communicated by the Author. 
t+ The @ priori demonstration shows the eases of failure. Suppose that 
the roots of a biquadratic equation are 1, 3, 5,9; then, givena+6=8, we 
know that either a=3, b=5, or else a=5, b=3, and in either case ad=15; 
hence in the present case (which represents the general case), a+b being 
known, the homotypical function ad is rationally determined. But if the roots 
are 1, 3, 5, 7 (where 1+7=3+5), then, given a+)=8, this is satisfied by 
3 aes 
G =). or by G = i and the conclusion is ad=15 or 7; so that here 
ab is determined, not as before, rationally, but by a quadratic equation. 
