[631 
631] SYNOPSIS OF THE THEORY OF EQUATIONS. 7 
e y n — 1=0, which 
roots; it is shown 
: a + /3i such that 
form cos 9 + г sin 9; 
or of the equivalent equations p x = Xa, &c., then 
a n — p x a n ~ 1 + ... =0, 
b n — pj) n ~ l + ... =0, 
Ve have thus the 
&c., 
(viz. it is for this reason that a, b, ... are said to be roots of x n — p x x n ~ l +... = 0); 
ilso obtain a like 
We are thus led 
and, moreover, that conversely from the last-mentioned equations, assuming that a, 6,... 
are all different, we deduce 
alued function, viz. 
is is 
p x = Xa, p 2 = Xab, &c., 
and 
x n —p x x lv ~ l + ... = (x — a) (x — b) .... 
denote the ?i-valued 
Observe that, if for instance a = b, then the two equations a n — p x a n ~ l +... = 0, 
b n — p x l') n ~ 1 + ... = 0 would reduce themselves to a single equation, which would not of 
itself express that a was a double root, that is, that (x — a) 2 was a factor of 
x n — p x x n ~ x + &c.; but by considering b as the limit of a + h, h indefinitely small, we 
refer here to the 
obtain a second equation 
na n-1 — (n — 1) p x a n ~ 2 + ... = 0, 
all which follows in 
ifications) applicable 
subject, it will be 
which, with the first, expresses that a is a double root; and then the whole system 
of equations leads, as before, to the equations p x = Xa, &c. But this in passing: the 
general case is when the roots are all unequal. 
We have then the theorem that every rational symmetrical function of the roots 
is a rational function of the coefficients; this is an easy consequence from the less 
general theorem, every rational and integral symmetrical function of the roots is a 
rational and integral function of the coefficients. 
In particular, the sums of powers Xa?, 2a 3 , &c., are rational and integral functions 
of the coefficients. 
зге are n quantities 
з they may become 
An ordinary process, as regards the expression of other functions Xa a b p , &c., in 
terms of the coefficients, is to make them depend on the functions Xa a , «Sic., but this 
is very objectionable; the true theory consists in showing that we have systems of 
equations 
p x = Xa, 
ing with the roots 
a, x — b,... in the 
laving the n roots 
f P'2 = 2a&, 
\ p x = Xa? + 2Xab, 
l Pz = Sa6c, 
\p x pz = Xa 2 b + 3 Xabc, 
le the existence of 
iven, then we write 
\ p x 3 = Xa 3 + 3Xa 3 b + 6Xabc, 
&c., &c. 
where, in each system, there are precisely as many equations as there are root-functions 
on the right-hand side, e.g. 3 equations and 3 functions Xabc, Xa 3 b, Xa 3 . Hence, in 
each system, the root-functions can be determined linearly in terms of the powers and 
products of the coefficients.