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stant; which equation is of the form (13). Suppose that the coefficient of Y, in (14) does not vanish ; and let equation;(14), for the sake of simplicity, be written,

B+B, Y + B, Y, + ...... + B, Y, = 0...........(15) In the same manner in which we proved B, to be an expression such as P, it can be shown that all the other terms, B, B ,, &c., are expressions such as P. Eliminate Y, from equation (15), as Y, was eliminated from (12). The result of the elimination is

BY d log

B Y + B, Y,

+ &c. = 0......(16)

dp As above, the coefficient of Y, here is an expression such as P. Also, if that coefficient vanish, we have B, Y, = k B, Yz, k being a constant quantity. And this equation is of the form (13). Should the coefficient of Y, in (16) not vanish, we may proceed to eliminate another of the terms, Y,, Yg, ...., Y.; and it will be found that the coefficient of Y in the equations that result from such eliminations can never at any stage become zero, unless such an equation as (13) subsist. Suppose then that all the terms, Y,, Y ,, ......,

..., Y. can be eliminated in the manner described, without the coefficient of Y, at any stage becoming zero. Then ultimately we get

HA + KAY, = 0, where I and K, the latter (and consequently also the former) not zero, are expressions such as P. And this is an equation of the form (13), m being taken equal to zero. Hence an equation such as (13) must necessarily subsist.

PROPOSITION II. In f (p), an integral function of a variable p, in a simple form, satisfying the conditions of Def. 8, let Y be a surd which is not subordinate to any other in the function, its index being 1. Arrange f(p) as follows:

f(0) = A + A. Yo + A,Y® + AY + &c., where A., A., &c., are expressions distinct from zero, and clear of the surd Y; A being also clear of the surd Y; and Yo, Yo, &c., being

Vol. V.

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D

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distinct powers of Y, not exceeding the (8-1)th. Let the surd T be a chief (see Def. 4) suburdinate of Y, but not a subordinate of any other surd in f (); its index being r; and, by changing T, wherever it occurs in f (p) in any of its powers, into zT, z being an goth root of unity, distinct from unity, let f (P), A, Y, A., &c., be transformed into F (p), B, U, B., &c.; so that

F (p) = B + B. U + B,U” + &c. Then, if F (p=f (p), the terms,

A, A,Y, A, Yo, &c.,... (1) taken in same order, are equal to the terms,

B, B.U , B, U , &c.,............... (2) each to each ; A being equal to B.

For, since F(p) = f (p), we have (A – B) + A, 8° + A,Y" + &c.--B. U— &c. = ................. (3) Hence (Cor. Prop. I) one or other of the following equations must subsist:

A_Y =D(A - B),
A.Y = DA. Y

.(4)
A.8° = DB.U

; where D is an expression involving only such surds as occur in the expressions A, B, A, B., &c., or are subordinates of Y or of U; Y being a term in the series, Yo, yo, &c, distinct from Yo ; and U representing some term in the series, U', u', &c. But, since T is not a subordinate of any surd in f (P) except Y, the coefficients B, B., &c., involve no surds different from those which enter into the coefficients A, A., &c.; and therefore involve only surds which are found in f (p). Also, since T is not subordinate to any of the subordinates of Y, it follows that the subordinates of U are the same with those of Y. Hence D involves only such surds as occur in f(p). Therefore (Cor. 1, Def. 9,) the first and second of equations (4) are inadmissible; and the third must subsist. Adopting then the equation, A, Y = D B.U”, we say that no other term in (1) than A. Y can be equal to the product of B, U by an expression such as

m

n

m

n

m

- D Bm

D; for, should A, Y = D, B.U", where D, involves only such surds, exclusive of Y, as occur in f (p), this would give us, D, A,Y* = DA, Y.

(5) Now D cannot be zero, else A. Y would vanish ; but A, is (by hypothesis) not zero; and the equation, Y = 0, is impossible by Def. 9. Hence, since D is not zero, equation (5) is (Cor. 1, Def. 9) inadmissible. Therefore we cannot have A, Y =

U. Consequently, as we established the third of equations (4), we can establish similar equations for all the terms in (1), after the first :

A, Y =D, B, U',

AnY" = D, BU, and so on ; the terms, A, Y, A.Y", A.Y", &c., being all different from one another, on the one hand; and the terms, B. U", B, U', B.Ư, &c., being all different from one another, on the other hand. Hence equation (3) becomes,

(A– B) Y (1-D ) Ac+Y" (1-D,') A.+ &c. = 0; where (Cor. 1, Def. 9) the coefficients, A-B, A.(1-D), &c., vanish separately. That is, the terms in the series (1), taken in some order, are equal to those in the series (2), each to each; A being equal to B.

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PROPOSITION III.

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Let f (p) be an algebraical function of a variable p, in a simple form; and let Y., Y,, &c., certain surds, with the common index, one of them a subordinate of any of the others, be such that all their subordinates occur in f (p). Suppose that

λα λα

Yc = PY, Y2 ...... Ya, or, as the equation may be written, Y=P,

(1) where Y is merely a symbol used (for the sake of simplicity) to de

λ, λ, note the continued product of the expressions Y., Yī , Y,

Take ,

k Pu,

λ, Y ; and P is an expression involving only such surds as occur in f (p); the whole numbers i, 12, &c., being less than 8. the general expression which includes (see def. vi.) all the cognate functions of f (p); one of its particular forms, distinct from f (p), being 01. In passing from f (p) to $, let P and Y become respectively Q and y; and, in passing from * to 4,, let Q and y become respectively P, and yi. Then the equation, Y1

(2) subsists; k being an ghe root of unity.

Explanatory remark.-When we speak of Y becoming y in passing from f () to , we do not assume that the expression Y is present in f(p); but we mean that all the surds which occur in f (p), and are also found in Y, must, in order that Y may be transformed into Y, undergo the same changes which they require to suffer in order that f (P) may become p.

We proceed with the proof of the Proposition. In the first place, should P be zero, Yc = 0. Let Y, be of the form,

Yc =

(a + ai Si + a2 S2 + + An Sn) ; where the coefficients, a, a,, &c., are rational; and each of the terms S,, S, , &c., is either some power of an integral surd, or the continued product of several such powers; the expression, a + ai Si + &c., satisfying the conditions of Def. 8. Then, since Ye = 0, we have

a + ai Si + a2 S2 + &c. = 0. Now all the surds present in this equation, being subordinates of Yc , are (by hypothesis) surds occurring in f(p), a function in a simple form. Therefore (Cor. 1, Def. 9) the coefficients, a, ai , &c., vanish separately. But, if Y. be what Yc becomes in passing from f (p) to $, and Y be what Y, becomes in passing from 0 to $i, we have

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} Yo = = (a + az si +

+ an

sa)' where sı, &c., are what S,, &c., become in passing from f (p) to di: Therefore Y=0. But, in the same way in which, from the fact that Yis zero, we have deduced the conclusion that Y is zero, we may, from the fact that P is zero, deduce the conclusion that P, is

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zero. Also, since Y. is a factor of y1, must be zero. Therefore y = k P1 .

In the next place, should P not be zero, the expressions y', Y , yi, developed by the ordinary process of involution, rendered integral, and made to satisfy the conditions of Def. 8, are of the forms,

y' A + A, v + A, t + &c.,
Y* = A + A, V + A, T + &c.,

(3) yi = A + A, V, + A, T,+ &c. ; where A, A1, &c., are rational; and each of the expressions, v, t, &c., is either some power of an integral surd, or the continued product of several such powers; the expressions V, T, &c., being what v, t, &c., become in passing from $ to f (p); and Vi, Ti, &c., what v, t, &c., become in passing from * to pi. In like manner, the expressions, Q; P', P, satisfying the conditions of Def. 8, are of the forms,

Q" B + B, m + B2 1 + &c.,
P = B + B, M + B2 L + &c.,

(4) P,= B + B, M,+ B2 Li+ &c.; where B, B, &c., are rational; and each of the expressions, m, I, &c., is either some power of an integral surd, or the continued product of several such powers ; M, L, &c., being what m, l, &c., become in passing from 0 to f (p); and M , L , &c., what m, 1, &c., become in passing from 0 to 01. From (1), (3), and (4), we have,

A + A, V + A2 T + &c. B + B, M + B2 L + &c...... (5) But the surds occurring in the expression on the left hand side of this equation, being necessarily subordinates of some of the surds, Y, Y., ......, Y, are all present in f (p). Those occurring in the expression on the right hand side of the equation are likewise all present in f (P). Therefore, since equation (5) subsists, the surd parts, V, T, &c., are (Cor. 2, Def. 9) severally identical, taken in some order, with the surd parts, L, M, &c. ; which also (Cor. 1, Def. 9) implies, that, if V be the surd part identical with M, A, is equal to Bi; and so on. But since V is identical with M, and Aj equal to Bi , and T is identical with (we may suppose) L, and A2 equal to B2 , and so on, the equation,

A + A, V1 + A, T, + &c. B + B, M; + B2 Li + &c, (6)

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