and so on, the expressions S1, S2, &c., are (by what we have proved) rational. But, by Newton's Theorem for the sums of the powers of the roots of an equation, (see equation (2),

S1+ A10.

Therefore A1 is rational

S2+ A1 S1 + 2 A2 = 0.

Therefore A2 is rational.

And in the same way all the terms A,, A,, &c., may be exhibited as rational expressions.

Cor. 1.-Should the terms in (1) not be all unequal, let the unequal terms be,

(5)

Then if f(p) be in a simple form, and X1 be the continued product of the terms, x—41, x—da, x-c, where 1, Pa, ...3 Pc, are a number of terms in (1), fewer than s, X, cannot have the coefficients of the various powers of a rational. For suppose, if possible, that powers of a rational. Then

X1 has the coefficients of the various 1 is a root of the equation, X1 = 0. And since, by the hypothesis made in the Corollary, ƒ (p) is in a simple form, 1 also (Prop. IV.) is in a simple form. Therefore (Prop. V.) all the terms in (5) are roots of the equation, X1 = 0; and they are all unequal: which, since the equation is of a degree lower than the sth, is impossible. Therefore X1 cannot have the coefficients of the powers of æ rational.

X1

фа

the coefficients, b1, b2, &c., may be exhibited as rational expressions; and, if ƒ (p) be in a simple form, each of the terias, 41, 42, recurs in (1) the same number of times. For let 1 occur A times in (1); 2, ẞtimes; and so on. Then

λ

β

^ ̊ —

(x−41) (x−2 ) ...(x—Ps) = (x−1 ) (x — P2 )... (x—m)=X........... (6)

The equation, X = 0, has one group of A equal roots, another group of ẞ equal roots, and so on. There is therefore a common measure,

The expression X2 resembles X in having the coefficients of the

dx

various powers of a rational; for it is the H. C. M. of X and 1·

X

Hence, denoting by Xs, we have, from (6) and (7),

X2

where Xs, being the quotient of X by X2, must have the coefficients of the various powers of a rational. Hence b1, b2, &c, may be exhibited as rational expressions. Thus the former of the two points to be proved in the Corollary is established. Next, shouldƒ (p) be in a simple form, and should the numbers A, B, &c., not be all equal to one another, let A be less than 8, and not greater than any of the others. Then, from (8) and (6), we have, putting X, to denote the λ

quotient of X by Xs,

X, being rational. Should the numbers, B-λ, 8-A, &c., not be all equal to one another, then, exactly as we reduced equation (6) to equation (9), on the left hand side of which no power of (x-1) appears as a factor, we can reduce equation (9) to an equation bearing the same relation to (9) that (9) bears to (6). And so on, till we arrive at an equation, such as (9), in which the indices, such as, ẞ-λ, &c., are all equal to one another. Let the result obtained when this point is reached be,

l-h

k-h

(x-a) (x-c) ......(x-8)

8-h

X5.

From this, since the numbers, l, k, ......, 8, are equal to one another, we get, by continuing the reduction,

(x-Pa) (x-c)...... (xg) = X6;

X being a rational expression: which, since the number of its factors, -a, -c, &c., is less than s, and since ƒ (p) is supposed to be in a simple form, is (Cor. 1) impossible. Hence λ, ß, &c., in (6), are all equal to one another; and therefore each of the terms, 1, P2, ....... Ps, must recur in (1) the same number of times.

Cor. 3. In f(p) let certain surds, y1, y2, &c., (in which series of terms, as was pointed out in Def. 7, all the subordinates of any surd mentioned are included), have definite values assigned to them; and

let the cognate functions of ƒ (p), obtained without departing from such definite values, (obtained, in other words, by proceeding without reference to the surd character of y1, y2, &c.,) be,

Then if

(10)

(x−1)(x —P2)...(x-n) = x2 + B1 x-1+ B2 "-2

+ &c., the coefficients, B1, B2, are equal to expressions which are rational as respects all surds except y1, y2, &c. In other words, no surds not included in the series yi, y2, &c., enter into these coefficients. The proof is the same as in the Proposition.

Cor. 4. In the case supposed in the preceding Corollary, it may be shown, as in Cor. 1, that, if the unequal terms in (10), (the definite values of y1, y2, &c., being understood to be adhered to), be,

and if ƒ (p) be in a simple form, and we write

(x-1) (x-Pa).........(x-c) = X1,

where the number of terms, 41, Pa‚........ ...., Pc, is less than t, these terms being terms in (10), X1 cannot involve, in the coefficients of the powers of x, merely the surds y1, y2, &c. For, if X1 did involve merely these surds, 1 would be a root of the equation, X1 = 0; and therefore (Cor. Prop. V.) all the expressions, 41, 42,................, 中 would be roots of that equation; the definite values given to yi, Y2, &c., being adhered to in all the expressions, 41, 2,..., . But these expressions are, by hypothesis, unequal. Therefore the equation, X1 = 0, has t unequal roots: which, since the equation is of a degree lower than the tth, is impossible. Therefore X1 cannot involve, in the coefficients of the powers of x, merely the surds y1, y2, &c. Cor. 5. In the case supposed in Cor. 3, let the unequal terms in the series (10), be, 1, 2,........., t; and let

t-2

(x-1) (X-Pg)......(x —pt ) = x2 +b1 x2-1 + bq x2 + &c.

--

Then the coefficients by, bg, &c., are equal to expressions involving no surds which do not occur in the series yi, ya, &c.; and, if ƒ (p) be in a simple form, each of the unequal terms, 41, 42 Pt, recurs the same number of times in (10) The proof is the same as in Cor. 2.

Cor. 6. If the equation, F (x) = 0, be an equation in which the coefficients of the powers of x are rational functions of p; and if F(x) cannot be broken into rational factors, (by which expression we mean, factors having the coefficients of the powers of rational), then, f (p), an integral function of p, in a simple form, being a root of the equation, F(x) = 0, the roots of that equation are identical with the terms of the series (5), that is, with the unequal cognate functions of ƒ (p). For (Prop. V.) every term in (5) is a root of the equation, F(x) = 0. Also (Cor. 2) the expression,

(x-1) (x-2)...

(x),..

(11)

when multiplied out, and arranged according to the powers of x, has the coefficients of the powers of equal to rational expressions. Therefore, unless the expression (11) were identical with F(x), F (x) would have a rational factor, of less dimensions, as respects x, than F(x) which is contrary to supposition. Therefore the expression in (11) is identical with F(x); and the roots of the equation, F(x) = 0, are the terms in the series (5).

PROPOSITION VII.

Let f (p) be an integral function of a variable p, in a simple form. Denote by 1, 2,......, Pn, all the unequal cognate functions of f(p), obtained by assigning definite values to certain surds, y1, y2, &c., and proceeding (according to Def. 7) without reference to the surd character of y1, y2, &c. Let

the coefficients A1, A2, &c., satisfying the conditions of Def. 8, and not involving (Cor. 5, Prop. VI.) any surds not found in the series, y1 y2, &c. Let y be a surd occurring in F1 (x), that is, in the coefficients, A1, A2, &c., but not a subordinate of any surd in F(x), its index being; and, when we substitute for y1 in A1, A2, &c., the successive values, zy1, 22 y1,....... z-ly1, z being an ɲth root of unity, distinct from unity, let F1 (x) become in succession F2 (x). Fs (x), &c. Then, if

F=F1(x) F2 (x) × F3 (x) x ......... x Fr (x)

= (x−1 ) (x−2 )......(x-n) (x-1+n)...(x-2)... (x-nr),

the terms, 1, 2,..., pnr, are all the unequal cognate functions of ƒ (p), obtained by giving definite values to all the surds in ƒ (p) which are present in the coefficients of the powers of x in F, and forming the cognate functions without reference to the surd character of the surds thus rendered definite: F being understood to be generated directly by the multiplication together of the factors, F1 (x), F2 (x), &c., and to have the coefficients of the various powers of x arranged so as to satisfy the conditions of Def. 8.

For, all the terms in the series,

P1, P2,........., Pn.......

(1)

are (by hypothesis) unequal. Suppose, if possible, that the terms,

which are the roots of the equation, F2 (x) = 0, are not all unequal. Then, F2 (x), having equal factors, has a measure, H, of less dimensions, as respects x, than F2 (x), and yet involving, in the coefficients of the powers of x, merely such surds as occur in F2 (x). But the surds in F2 (x) are identical with those in F1 (x). [For instance, let

1

F1 (∞) = (1 + √/p)3, and, F2 (x)=(1+ √/p), where z is a third

x

root of unity, distinct from unity. The presence of z in F2 (x) does not affect the surds in the expression]. Therefore the expression H, of less dimensions as respects x than F1 (x), involves in the coefficients of the powers of merely such surds as appear in F1 (x): which, [since F1 (x) is the product of the terms, x-P1,..., Pn, where 1, 2,.... Pn, are all the unequal cognate functions of ƒ (p) obtained by assigning definite values to certain surds in ƒ (p)], is (Cor. 4, Prop. VI.) impossible. Therefore all the terms in (2) are unequal. Next suppose, if possible, that some term in (2) is equal to a term in (1). Then F2 (x) and F1 (x) have a common measure; and their H. C. M. involves only such surds as appear in F1 (x) or Fg (x); that is, only such as appear in F1 (x): which, as above, is (Cor. 4, Prop. VI.) impossible, unless F1 (x) and F2 (x) are identical. Suppose then, if possible, that F1 (x)= F2 (x). The coefficients of like powers of a must be equal. Let the coefficient of a certain powers of x in F1 (x), arranged according to the powers of y1, (we choose a coefficient where y1 occurs in some of its powers), and satisfying (as, by hypothesis, it does) the conditions of Def. 8, be