A “SMITH’S PRIZE” PAPERO; SOLUTIONS. 
[From the Oxford, Cambridge and Dublin Messenger of Mathematics, vol v. (1870), 
pp. 182—203.] 
1. Mention what form of given relation <j> (a, b, c, ...) = 0 between the roots of a 
given equation will in general serve for the rational determination of the roots; explain 
the case of failure; and state what information as to the roots is furnished by a given 
relation not of the form in question. 
In the given relation, (a, b, c,...) must be a wholly unsymmetrical function of 
the roots; that is, a function altered by any permutation whatever of the roots; or, 
what is the same thing, by any interchange whatever of two roots. 
For this being so, if a, /3, 7,... be the values of the roots, then for some one 
order, say a, ¡3, 7,..., of these values the given relation <£ (a, b, c, ...) = 0 will be satisfied 
by writing therein a = a, b — /3, c = 7, &c.; but it will in general be satisfied for this 
order only, and not for any other order whatever (viz. it will not be satisfied by 
writing a = /3, b = a, c = 7, &c., or by any other such system). The given equation 
determines that the roots are equal to a, /3, 7, ... in some order or other, but the given 
equation combined with the given relation <£ (a, b, c, ...) = 0, determines that a is = a 
and not equal to any other value, b = ¡3 and not equal to any other value, &c.; and 
it thus appears a priori, that the two together must rationally determine each of the 
roots a, b, c,...; the a posteriori verification, and actual rational determination of the 
values of a, b, c,... respectively, is a separate question which is not here considered. 
The function (f)(a, b, c,...) may be of the proper form, and yet the particular 
values a, /3, 7,... be such that the given relation 0 (a, b, <?,...) = 0 is satisfied, not only 
for the single arrangement a —a, b = /3, c = y, &c., but for some other arrangement, 
1 Set by me for the Master of Trinity, Feb. 3, 1870.