one of these is 0, so that 6 n = a, then, assuming x — 6y, we have y n —1=0, which 
equation (like the more general one x n — a = 0) has precisely n roots; it is shown 
that these are 1, g>, eo 2 ,.., &> n_1 , where eo is a complex number a + /3i such that 
a 2 + /3 2 = l, or, what is the same thing, a complex number of the form cos 0 + sin 0; 
2?r 
and it then at once appears that 0 may be taken = 
We have thus the 
trigonometrical solution of the equation x n — 1 = 0. We may also obtain a like 
trigonometrical solution of the first-mentioned equation x n — a = 0. We are thus led 
to the notion (a numerical) of the radical a n , regarded as an w-valued function, viz. 
any one of these being denoted by {/(a), then the series of values is 
y/(«), cojZ(a),.., co n_1 (a). 
Or we may, if we please, use y/(a), instead of a n , as a symbol to denote the ?i-valued 
function. 
It is not necessary, as regards the equation x n — 1 = 0, to refer here to the 
distinctions between the cases n a prime, and a composite, number. 
As the coefficients of an algebraical equation may be numerical, all which follows in 
regard to algebraical equations, is (with, it may be, some few modifications) applicable 
to numerical equations; and hence, concluding for the present this subject, it will be 
convenient to pass on to algebraical equations. 
We consider, secondly, an algebraical equation 
- p^™* 1 + ... = 0, 
and we here assume the existence of roots, viz. we assume that there are n quantities 
a, b, c,... (in general, all of them different, but in particular cases they may become 
equal in sets in any manner), such that 
x n —p-ipc 11-1 + ... = (x — a) (x — b) .... 
Or, looking at the question in a different point of view, and starting with the roots 
a, b, c,... as given, we express the product of the n factors x — a, x — b,... in the 
foregoing form, and thus arrive at an equation of the order n having the n roots 
a, b, c,.... In either case, we have 
Pi = 2a, p 2 = Xab,.., p n = abc..., 
viz. regarding the coefficients p 1 , p 2> .., p n as given, then we assume the existence of 
roots a, b, c,... such that p 1 =l£a ) &c., or regarding the roots as given, then we write 
p u p 2 , &c., to denote the functions Xa, Xab, &c. 
It is to be noticed that, in virtue of 
x n — p- i x n ~ 1 + ... = {x — a) (x — b), &c., 
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