pression L involves only such surds as occur in Fe (x) or fe (x); but, since U is (by hypothesis) not a subordinate of any surd in the function Fe(x), the substitution of 21 U for U makes no change on the surds appearing in the function: that is, the surds in fe (x), and therefore also those in fe (x), are identical with those in Fe (x); and consequently the surds in L are all found in Fe (x). Now the simple factors of F(x) are (see Prop. VIII.) the unequal cognate functions of ƒ (p) obtained by assigning definite values to those surds in ƒ (p) which are also present in Fe (x), and taking the cognate functions without reference to the surd character of the surds so made definite. Therefore (Cor 4, Prop. VI.) no expression such as L can involve merely such surds as appear in Fc (x). Hence Fe (x) cannot but be

a

equal to some term in (5). Let F. (x) = "ƒe (x).

fe (x). This implies that the coefficients of like powers of x in these expressions are equal.

E

Let x be a power of x in Fe (x) involving in its coefficient the surd

th

Y in one of its powers; the coefficients, D and D1, of the E power of x in Fe (x) and fe (7) respectively being,

a

where such terms as b1 are clear of the surds Y and U, and not zero;

k

U'

and no two terms such as that written Y are identical; k not being zero. Since s is a prime number, and [Fc (x) satisfying the conditions of Def. 8] k is less than 8, we can find whole numbers, w and w1, less than s, and such that

Now (Y)", when expressed as an integral function satisfying the conditions of Def. 8, involves only the subordinate surds of Y. Therefore, by the equation found, we can eliminate Y from Fe (x), introducing in its room powers of V, but no powers of any other surd that was not previously in the function. Let F. (x), as thus exhibited, be written Fe (x). A term in Fe (x) is b1 V U'. Should

с

be neither zero nor unity, write V1 for U; and, as above, all the powers of U occurring in F. (2)can be made to disappear; powers of V1 being introduced into the function in their room, but no powers of any other surd that was not previously in the function. Let the function become, in consequence of this change, Fe (x); and let the coefficients of the several powers of x in Fe(x) and Fe (*) be supposed to satisfy the conditions of Def. 8, as was the case with Fe (x). Then, since V and V1 are respectively powers of surds that were present in F. (x), but do not remain (except as implicitly involved in V and V1) in F′′c (x); and since Fc (x) is (by hypothesis) in a simple form, Fe () and Fe() are also in a simple form. In changing

с

This may now

Fe (x) into Fe (x), we assumed that I was not zero. be shown to be case. Equate the coefficients of E in Fe (x) and f(); this latter expression being what fe (x) becomes when Y is eliminated, and powers of V introduced in its room. Then,

+ b1 VU+

=

a

Ik (a-1) 1

+b VU z 21+ &c.

........+b1 {1--2k (a-1) z } V U1 &c. = 0.

Therefore, by Cor. 1. Def. 9, 1--z

k (a-1) 1
21=0

(6)

If I were zero, this would make z(a-1) equal to unity: which, since the numbers, k, a-1, are less than s, and z is an sth root of unity, distinct from unity, is impossible. Therefore l is not zero; and hence F. (x) can be exhibited in the form F. (x). Equate the coefficients of x in F. (x) and f (x); this latter expression being

E

"

what ƒe (x) becomes when Y and U are eliminated, and V and V1 are introduced in their room. Then the equation (6) still holds. But such an equation implies, that, z being an sth root of unity, and z1 being a σth root of unity, distinct from unity, σ=s. Again, suppose V to occur in its h power in any term of Fe (x), so that b2 is a term in the coefficient of some power of x. Then, by reasoning as above, we get

I

"

Hence, by a comparison of (6) and (7),

(h-h1)

1-21

= 0. h=h1 :. V v1.

1 =

Hence b2 VV1 becomes b2 V V1: which, again, by returning

from V and Vito Y and U, becomes 6 Y

kh-wsh-w1; where ws is

the greatest multiple of s in kh, and was is the greatest multiple of sin lh; b being an expression clear of the surds Y and U. Consequently Fe (x) may be written,

B being an expression clear of the surds Y and U; and the numbers, k, l, remaining the same in all the terms, such as BY kh-ws-w1, U included under the symbol . But equation (8) implies that the surds Y and U are similarly involved in Fc (x).

PROPOSITION X.

Let f (p) be an integral function of a variable p, in a simple form, satisfying the conditions of Def. 8; and let

where Y is a surd in ƒ (p), with the index; and A is an expression, not zero, involving only surds, distinct from Y, which occur in f(p); A being a whole number, not zero, and less than 8; and the expressions B, U, are what A and Y respectively become on changing T, a chief subordinate of Y, with the index, into z T, z being a oth root of unity, distinct from unity; T not being a subordinate of any surd in the expression A. Then the surd Y is of the form,

where H is an expression clear of the surd T; and m is a whole number, less than σ. Also, σ is not equal to 8.

For, let be the general expression which includes all the cognate functions of A Y, taken without reference to the surd character of any of the surds in A Y^, except T and Y; and let ', arranged so as to satisfy the conditions of Def. 8, be,

where is an indefinite oth root of unity; and D, D1, &c., are clear

2

2

.. T D1 (1 − 2) + T2 D2 (1 − z2 ) + &c. = 0.

This equation involves only such surds as occur in f(p). Therefore (Cor. 1, Def. 9) the coefficients, D1 (1-2), D2 (1-2), &c., vanish separately. But, since σ is a prime number, and z is a oth root of unity, distinct from unity, none of the terms, 1 z, 1-2, &c., vanish. Therefore D1, D2, &c., must all vanish ;

and

Raise both sides of this equation to the 7th power; r and ʼn being whole numbers such that

where P and Q involve only such surds, exclusive of Y, as are present

in A Y; and Q is clear of the surd T.

2

Let the forms of P and Y be,

Р

b + b1 T + b2 T +

(4)

(5)

where b, b1, &c., h, h1, &c., are clear of the surd T. Suppose, if possible, that the terms b1, b2, &c., are all zero. Then P = b; and, b* h + b* h2 T + b′ h2 T2 + &c. = Q.

8

S

But since Q is clear of the surd T, the coefficients, b h1, b h2,&c., in this equation, must (Cor. 1, Def. 9) vanish separately. Now, A is (by hypothesis) not zero; therefore P is not zero; therefore b is not

zero.

Therefore all the terms, ha, ha, &c., vanish: which (since T

is a subordinate of Y) is impossible. Hence at least one of the terms, b1, bg, &c., as be, does not vanish. But this leads to the conclusion

except be, must vanish. For, from (3) and (4), we have,

From this equation eliminate the surd Q, in the same way in which X1 was eliminated from equation (4) Prop. I. The result is,

+ be Y TR E + b, Y T" E1 + &c. = 0.

The conditions necessary in order that E and E1 may both vanish, are,

k and q being constant quantities; and these equations give us, q be T = k by T":

which, T and T being distinct powers of T, not exceeding the (-1)th, is (Cor. 1, Def. 9) impossible. Therefore b, is the only term in (6) which does not vanish; and, from (3), (4), and (5),

If hm be a term in the series, h, other terms in that series vanish.

h1, &c., which is not zero, all the For, if he be another term, let

C 8+ e = w σ + ß,

and, csm = 01 σ + δ;

where ẞ and are whole numbers, less than σ. Then, since e and m are not equal, and each of them is less than σ, ẞ and 8 are not equal. And so likewise as regards the other terms. Therefore (Cor. 1, Def. 9), all the coefficients, be h, bh, &c., in (7), must vanish, except the one occurring in the term which is equal to Q. But hm does not vanish. Therefore all the terms, h, hy, &c., except hm, or (as we may write it) H, must vanish; and Y is reduced to the form,

Y = (H TTM).