50 
[310 
310. 
NOTE ON ME JEEEAED’S EESEAECHES ON THE EQUATION 
OF THE FIFTH OEDEE. 
[From the Philosophical Magazine, vol. xxi. (1861), pp. 210—214] 
Functions of the same set of quantities which are, by any substitution whatever, 
simultaneously altered or simultaneously unaltered, may be called homotypical. Thus all 
symmetric functions of the same set of quantities are homotypical: (x + y — z — icf and 
xy + zw are homotypical, &c. 
It is one of the most beautiful of Lagrange’s discoveries in the theory of equations, 
that, given the value of any function of the roots, the value of any homotypical 
function may be rationally determined 1 ; in other words, that any homotypical function 
whatever 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 Reso 
lution 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 order in W; and 
he conceives that he has shown that one of the roots of this equation 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.” 
1 The a priori demonstration shows the cases of failure. Suppose that the roots of a biquadratic equation 
are 1, 3, 5, 9; then, given a+ 6 =8, we know that either a=3, 6=5, or else a = 5, 6=3, and in either case 
ab = 15; hence in the present case (which represents the general case), a + 6 being known, the homotypical 
function ab is rationally determined. But if the roots are 1, 3, 5, 7 (where 1 + 7 = 3+ 5), then, given a + 6 = 8, 
this is satisfied by (a, 6 = 3, 5) or by (a, 6 = 1, 7), and the conclusion is a6 = 15 or 7; so that here ab is deter 
mined, not as before, rationally, but by a quadratic equation.