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which (Cor. 1, Def. 9) is only possible if the number, cs+m, be a multiple of o. Hence σ and s are not equal to one another.

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Cor. If an equation such as (3) subsist, the form of the surd Y is that given in (2), and s and σ are unequal.

PROPOSITION XI.

In the series, in Prop. VIII.,

xf (p), F1 (x),

......

Fa (x) or X,

let the factors by whose continued product Fe+1 (x) is generated, be,

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с

(1)

c+1

where Fe (), Fc (x), &c., are what Fe () becomes, on substituting successively for Y, a surd in Fe (*), not subordinate to any other surd in the function, and having the index, the values z Y, 22 Y, &c.; ≈ being an sth root of unity, distinct from unity. Also suppose, that, besides Y, there are no surds in Fe (x) which disappear from F (x), except subordinates of Y. Let T, a chief subordinate of Y, with the index, disappear from Fe+1 (), in which case T is not a subordinate of any surd in Fe (x) except Y. When T is changed into z, T, 21 being a σth root of unity, distinct from unity, let the terms in (1) be transformed into,

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Generally, if X be what F. (x) becomes when T is changed into z T, b being a whole number in the series, 1, 2, ....., σ—1, the expression, X, comprehends 8 (0-1) particular forms:

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Then, if Fe (x) be equal to a term in (2), an equation,

Y = PY1,

(3)

must subsist; where Y1 is what Y becomes when T is replaced by z1 T ;

and P is an expression involving only surds which occur in Fe (x), exclusive of Y; A being a whole number, distinct from zero and less than s, satisfying the condition,

σ

= w s + 1 ;. . . . .

(4)

where w is a whole number. Also, if Y be not of the form shown in equation (2) Prop. X, A is not uuity, and s is not 2.

Let us in the mean time reason on the supposition that Fe (x) is equal to a term in the first horizontal line of (2). We here make an observation to which we shall have occasion subsequently to refer. When an expression is equal to some term in a series such as that constituted by the terms in the first horizontal line of (2), any one of the terms in the series may be assumed to be that to which the expression in question is equal; because any particular term in the series stands, in fact, as the representative of all the terms in the series, in consequence of the s distinct values which may be given to the surd Y1 Proceeding, therefore, on the supposition that Fe (~) is equal to a term in the first horizontal line of (2), we may understand that Fe (x) is equal to 1X1. Take x, a power of x in Fe (x) having some power of Y present in its coefficient; and let the coefficient of x in Fe (x), satisfying the conditions of Def. 8, be,

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where Ac. An, &c., none of them zero, are clear of the surd Y; no two powers in the series, Y, Y", &c., being identical. Then, D being the corresponding coefficient in 1X1, and Be, Bn, &c., being what Ac, An, &c., bcome when T is changed into 1 T, we have

But D= D1.

taken in some

each to each

m

=

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+ Be Y1 + Bn Y1 + &c.

1

1

D1 Therefore, by Prop. II., the terms, A, Y, AY", &c., order, are equal to the terms, B. Y°,

Hence we may put

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where Bm Y is some term in the series, Be Y

1

B n

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n

Y, &c.,

By Y1, &c.

And, since A, and Bm involve only surds which occur in Fe (x), exclusive of Y, this equation can easily be reduced to one of the form (3); A not being zero, because neither c nor m is zero. Now, from equation (2), the following may be derived by Prop. III. :

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and so on; where k1, ką, &c., are sth roots of unity; and P1, P2, &c., are what P becomes when T is successively changed into 21 T, z T, &c.; Y1, Y2, &c., being what Y becomes when T is successively changed into z T, z T, &c. By eliminating Y1 betwixt equation (3) and the first of equations (5), Y1 and Y. betwixt (3) and the two first of equations (5), and so on, we get

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where n is any whole number whatsoever; and p is the greatest multiple of in A"; and Q is an expression which involves only such surds as occur in F. (*), exclusive of Y; none of the surds which it involves having T as a subordinate. Now equation (7) has been found on the hypothesis that F. (x) is equal to a term in the first line of (2). But, by the same course of reasoning, an equation such as (7) may be established, should F. (x) be given equal to a term in any line of (2). And equation (7) includes the form (3). Therefore, when F. (x) is equal to a term in (2), whatever be the line of (2) in which that term occurs, an equation such as (3) subsists. In order to establish equation (4), we observe that equation (7), when a is taken equal to σ, becomes,

Y = Q Y

Let p -m, m being less than s. Then

Y — Q Y” — 0.

But, since F. (2) is a function in a simple form, this equation

m

is (Cor. 1, Def. 9) impossible, unless Y and Y be be the same power of Y, that is, unless m be unity. Therefore

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an equation of the form (4). Should be unity, it is plain, referring to the manner in which the first of equations (6) was obtained, that A. Yo — B ̧Y1 ;

с

and consequently (Prop. X.) the surd Y is of the form shown in (2) Prop. X.; so that, if Y be not of that form, A cannot be unity. In this case, also, s cannot be 2; for were s equal to 2, λ could have no other value than unity.

Cor. 1.—Should F. (2) not be equal to a term in (2), then no such equation as (3) admits of being formed. For, since T disappears from Fe+1(), the continued product of the terms in the first horizontal line of (2) is equal to that of the terms in (1): both products being F(x). (x). Hence F. (x) has a common measure with some term in the first line of (2), which term (on the principle pointed out in the Proposition) may be assumed to be 1X1. Let L be the H. C. M. of Fe (x) and 1X1. Since the roots of the equation, F(x) = 0, are (Prop. VII.) the unequal cognate functions of ƒ (p), obtained by assigning definite values to those surds in ƒ (p) which are also present in F, (x), and taking the cognate functions without reference to the surd character of the surds so rendered definite, L, which is of less dimensions, as respects x, than F. (x), cannot (Cor. 4. Prop. VI.) involve, in the coefficients of the powers of x, merely such surds as occur in F. (x). But the only surd not in F. (x), which can possibly appear in L, is Y1; because, with the exception of Y1, all the surds in 1X1 are found in F. (x). Hence Y1 cannot be absent from L. But if such an equation as (3) subsisted, all the powers of Y1 in L might be eliminated from L, without any surds being introduced into L, except such as are found in F, (x). Hence no such equation as (3) can be formed.

1

Cor. 2.-Should no equation such as (3) subsist, any function involving merely such surds as are in F. (a), together with Y1, is in a simple form. For suppose, if possible, that ↓ (p) is such a function, and that it is not in a simple form. Then an equation such as (1) Prop. I. must subsist; all the surds occurring in it being found in (p). One of these must be Y1; else all the surds in the equation

would be present in F. (x): which is impossible. Also Y, is not a subordinate of any surd in the equation, because all the surds in the equation except Y1 are present in F. (x), and Y, is not in F. (x); so that no surd to which Y1 is subordinate can appear in F. (x). Let then the equation, satisfying the conditions of Def. 8, be,

λ λι

В В1

H+ H1 Y Y1 + H2 Y Y1 + &c. = 0,

where H, H1 &c., are clear of the surds Y and Y1; at least one number in the series, A1, B1, &c., (say λ), not being zero; the corresponding coefficient H1 being at the same time distinct from እ λι В В1

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zero; and no two terms in the series Y Y Y Y1, being identical. Then (Cor. 1, Prop. I.) an equation,

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must subsist; where P is an expression involving only such surds es occur in the expressions H, H1, &c., or are subordinates of the B B1

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surds Y, Y1; m being either unity or zero: the term Y Y standእ λι B 81 ing as the type of any term in the series, Y Y1 Y Y, &c., after the first. But, should m be zero, equation (8) is of the form (3): which, since F. (x) is not equal to a term in (2), is (Cor. 1) inadmissible. Should m be unity, equation (8) becomes

с

X-B A-B1
Y Yi = P.

Here, by hypothesis, the numbers A-B, A-B1, do not both vanish. Should the latter vanish, the equation is at variance with the supposition that F. (x) is in a simple form. Should the former vanish, the equation is at variance with the fact that 1X1 is in a simple form; which, however, it must (Prop. IV.) needs be. Should neither vanish, the equation is of the inadmissible form (3). Hence the function(p) cannot but be in a simple form.

Cor. 3.-Should F. (x) not be equal to a term in (2), the equations,

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