534] 
A “ SMITHS PRIZE ” PAPER ; SOLUTIONS. 
475 
a = 3, b = 7, c = /3,... or for more than one such other arrangement. (For instance, if 
the given relation be a + 2b + 3c - 32 = 0, and the roots are 3, 5, 7; the relation is 
satisfied by a = 5, b — 3, c = 7, and also by a — S, b= 7, c = 5.) Here the given equation 
and relation do not completely determine each root, they only determine that a is = a 
or = 8 (or as the case may be = some other one value); and similarly that b is = ¡3 
or =7 (or as the case may be = some other one value), and so for the other roots 
c, d,...; and it thus appears a priori, that in such a case each root is determined, 
not rationally, but by means of an equation, the order of which is equal to the 
number of the values of such root; we have here the case of failure of the general 
theorem. 
When the given relation cf)(a, b, c, ...) = 0 is not of the required form; that is, 
when (f>(a, b, c,...) is a partially symmetrical function, there will be in general several 
arrangements of a, /3, 7,..., such that equating a, b, c,... to a, /3, 7,... according to 
each of these arrangements, the given relation ft (a, b, c, ...) = 0 will be satisfied; and 
it follows that each of the roots a, b, c, ... is determined not rationally, but by means 
of an equation of a certain order (not necessarily the same order for each of the 
roots). Thus, if the relation be symmetrical as regards a pair of roots a and b; then 
if it be satisfied, suppose by a = cl, b = ¡3, c = 7,..., it will also be satisfied by a = /3, 
b — a, c = 7,..., but not in general in any other manner; each of the roots a, b has 
here either of the values cl, ¡3, and the two roots a, b in question will be given, not 
rationally, but by means of the same quadratic equation. And observe, moreover, that 
any other function (a, b, c,...) of the same form as <£, that is, symmetrical in regard 
to the two roots a, b, will for the two arrangements a = a, b = f3, c = 7 ..., and a = ¡3, 
b = ci, c = 7,... acquire not two distinct values, but one and the same value, that is, 
the value of yjr(a, b, c,...) will be determined rationally; and so in general. 
There is for the partially symmetrical function </> (a, b, c, ...) a case of failure 
similar to that which arises for the completely unsymmetrical function, viz. the particular 
values cl, ¡3, 7 ... may be such as to give more ways of satisfying the given relation 
cf)(a, b, c, ...)= 0, than there would be but for such particular values of a, ¡3, 7,...; 
and there is then a corresponding elevation of the order of the equation for the 
determination of the roots a, b, c, ... or some of them. 
2. If the roots (a, ¡3, 7, 8) of the equation 
(a, b, c, d, e)(u, 1) 4 = 0 
are no two of them equal; and if there exist unequal magnitudes 6, </> such that 
We have 
and we cannot have 
other; for this would 
J J <f> + a 0 + /3 
imply (6 — (f)) (a — /3) = 0, that is, 6 = <fi, or else a = /3. 
60—2