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and subordinate being both reckoned), will be less, by at least one, than it was before. Again, should the function involve a surd Y, of the form shown in (1), then, since 8 and o are unequal prime numbers, we may choose c and w, whole numbers, such that

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Let HT when made to satisfy theconditions of Def. 8, be written K; and substitute for Y, wherever it occurs in the function in any of its powers, the value furnished by the equation,

Y=T*(K)

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where it will be observed that the surd Kohas no subordinates which were not subordinates of Y, while it has not as a subordinate the surd T, which was a subordinate of Y.

Once more, suppose that two surds, V and V, with the common index 1 similarly involved in the function : that is to say, when the function has been arranged according to Def. 8, wherever one of the surds V and V, appears in any of its powers, it occurs multiplied by a

λ,
BL

8, power of the other ; as V by Vi, v by Vi V by Vi i and so on ; the pairs of equations (3) and (4). Prop. IX, subsisting. Let V=u®, and V,=0> ; and put

λ

B

8

1

u

λι

λι

V Vi

(

U U

h

Y. by hλι ks+B

q8+B1
:: V V

Vi
B BI

:. V V AY' A being put for v-ks v-48.

Since the surds V and V, have the common index, the expression A may be exhibited so as to involve only surds which are subordinates of V or Vi. Let A be so ex. hibited. In like manner,

8 81
V V - BY

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Н

1

h

н

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where B is an expression of the same character with A. And so on.

λ λι B B1 8 81 Substitute for V V V Vi, V Vi, the values, Y , AY , BY &c. Then, when the function is rendered integral, the number of surds present in it, (principal and subordinate being both reckoned), will be less, by at least one, than it was before. Let modifica. tions of the three different kinds described continue to be made as far as possible. It is obvious that a limit will ultimately be reached ; and if the function be then rendered integral, it will be an integral function in a simple form, containing no surd such as Y in (1), and having no two surds similarly involved in it.

Cor.-In f (P), a function which has been made to undergo the modifications described in the Proposition, let Y be a surd, not subordinate to any other in the function ; its index being. Also, let T and t be two surds, with the common index as which is not equal to şi

, subordinate to Y, but neither of them subordinate to any other surd in f (p); and suppose that the form of Y is,

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............

Y = (HT")

.....(2) where m is a whole number, less than o ; and H is an expression in which the surds T and t are similarly involved. As in the Proposition, we can choose c and w, whole numbers, such that

CS + m = wo, :: Y=T“ (K)';

* where K is put for H TW"; that is, K is the product of an expression which is clear of the surds T and t, by one in which T and t are similarly involved. Hence again, as in the Proposition, we can eliminate the surds T and t from K, introducing in their room a single new surd V; one of the surds T and t, as t, disappearing from the function altogether. And, since T and t are not subordinates of any surd in f(p) except Y, the function, after being subjected to this change of form, may still, if necessary, be made to satisfy the different conditions described in the Proposition. So that, upon the whole, an algebraical function of a variable p may be exhibited as an integral function in a simple form, with no two surds similarly involved in it; nor with any surd involved in it of the form (1); nor with any surd involved in it, which, while not subordinate to any other in the function, is of the form (2).

PROPOSITION XIII.

Let f (p) be an integral function of a variable p, in a simple form, containing no surd such as Y in (1) Prop. XII, nor any surd, which, while not subordinate to any surd in the function, is of the form shown in (2) Prop. XII ; and having no two surds similarly involved in it. Let Y be a surd in f (P), with the index , not subordinate to any other in the function; and let the function, arranged so as to satisfy the conditions of Def. 8, be,

f (p) = A +A, Yo + A, Y" + &c.;

.... (1)

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where Ac, An, &c., none of them zero, are clear of the surd Y; A also being clear of the surd Y; and Yo, Y", &c., are distinct powers of Y. Suppose that T and T, are two chief subordinates of Y, with the indices

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but that neither of them is a subordinate of any other surd in the function f (p). When T is changed into z, T, z, being a oth root of unity, distinct from unity, let f (P), Y, A, A., &c., be transformed into fi(P), Y., B, B., &c.; and, when T, is changed into z,T,, 2, being a one root of unity, distinct from unity, let these same expressions become f2(P), Y, b, b, &c.; so that fiip) = B + B.Y; + B.Yo + &c.,

(2) and, f 2(P) = b + be(19°) +ba(, Y") + &c. Then, if fi(p)=f(p), it can be proved by the same reasoning as in Prop. II, that the terms,

B.Y; , B.Y", &c., taken in some order, are equal to the terms,

be (Y), b. (Y"), &c. But should the numbers o and o, not be both equal to s, the equation, B.Y; = be (77°),

(3) cannot subsist.

For suppose, if possible, that equation (3) subsists. Then the reasoning of Prop II. makes it plain that B.Y" is equal to b, (,Yo ), and so on.

We will therefore assume this. The gth powers of A. YR B.Yị, and be (,Y*), arranged so as to satisfy the conditions of Def. 8, are of the forms, (A.Y) =

D + D), TTM t; +D,TT, + &c., (B.Y;) = D + Z, D, T T, +2, D,T T, + &c., {b:(Y)} = D +2° D, "T, +2, D, T*T, + &c.,

M

N

m

M

M м.

N

where D, D,, &c., are clear of the snrds T and T, ; no two terms in the series, TT, TT , &c. being identical with one another. There must be at least one term in the series D,, Dą, &c.; else (A.Yo ) would be reduced to D; in which case (Cor. Prop. X.) the surd Y would be of the inadmissible form given in (1) Prop. XII. But, from (3) and (4), Tat" D, (27") + T T;" D, (2

q" t;* D, (2 – 2,3 + &c. = 0.

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Hence, by Cor. 1. Def. 9, the coefficients of T T,, TT, , &c., vanish separately. But, since the expressions on the right hand side of (4) satisfy the conditions of Def. 8, the terms D,, D,, &c., do not vanish. Therefore

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and so on : from which it follows that 0,0; and also that 2, -3; where u is a whole number, less than o, and such that

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}

W,0 + m,

(5) u N- w20 + M, and so on; m, wg, &c., being whole numbers. Let V be put for T T;. Since o, has been proved equal to o, the surds T and T, have a common index ; and V may be considered a surd with the same index as that of T and Tn. Take W and wg, whole numbers less than 0, and such that

Wm

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M

X

W

- HV ;

W

and, TT, w, being a whole number; and H, an expression clear of the surds T and T. And so on. Hence the form of (A.Y*)" is, (A,Y*) – D + D,V + D,H V + &c. ;

(6) neither of the surds, T, T,, appearing (except as implicitly involved in V) on the right hand side of the equation. From the expressions A, and Y let the surd T, be eliminated, by substituting for it, whereever it occurs in any of its powers, its value derived from the equation,

m n

V-T T;

(7) and, when thus modified in form, let A, and Y, satisfying the conditions of Def. 8, become respectively P and U. Then the surd T cannot (otherwise than as implicitly involved in V) be a subordinate of U. For suppose, if possible, that it is. Put Q to represent the expression, D + D, V + &c. Then

1

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PU

(8) Now, any function involving merely such surds as occur in equation

1 (8), exclusive of Q is in a simple form ; for, all the surds in the expressions, P, U, and Q, except the surd V, are found in f (P); and, if an equation such as (1) Prop. I. could be formed, involving the surd V, that equation, when V was replaced by TT', would be reduced to a corresponding equation involving only surds in f (P); which, since f (p) is in a simple form, is impossible. Hence, since the surd T, a chief subordinate (on the hypothesis at present made) of U, is not present in Q, it is (by Cor. Prop. X.) implied in equation (8) that the form of U is

U

- (LT) where L is an expression involving merely such surds, exclusive of U and T, as occur in the expressions P, U, and Q; and I is a whole number less than s. Restore U to the form Y; and let the surd V, in L, be replaced by its value in (a). Then

1

S

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