с

L1, that of F, (x) and 2X1; L2, that of F. (x) and 3X1; and so on; and K is the H. C. M. of F. (x) and 1X2; K1, that of Fe (x) and X.; and so on: all the expressions L, K, L1, K1, &c., being of the same dimensions as respects x. For, since F. (x) necessarily has a common measure with more than one term in the first line of (2), let us (on the principle pointed out in the Proposition) take ,X, to be a term in that line, such that the H. C. M. of F. (x) and ,X, is not of less dimensions, as respects, than the H. C. M. of F. (*) and any other term in the first line of (2). Then, X' being the general symbol under which all the terms in the first line of (2) are comprehended, let the H. C. M. of F, (x) and X' be sought in the ordinary method; the process being continued till that stage is reached, where, in the case of F. (*) and 1X1, the operation has an end. Let the remainder R, [that is, in the general case of F. () and X, reduced to an integral function, and satisfying the conditions of Def. 8, no two terms such as Y Y1 in the coefficient of any power of a being identical, be,

እ λι

૧ and the corresponding coefficients which are not expressed being clear of the surds Y and Y,. Then, if R, be what R becomes in the particular case of F. (x) and 1X1,

=

where z' is some (not definite) power of z. But Ri 0. This implies that the coefficients of the different powers of a vanish separately. Also (Cor. 3) any function involving merely such surds as are in F, (x), together with Y1, is in a simple form. Therefore,

if

(p) is a function in a simple form. Therefore (Cor. 1, Def. 9) q, with all other such coefficients, must be zero. Therefore R vanishes, as well as R1. And, in the case when F. (x) is compared with any one in particular of the terms in the first line of (2), it is not possible for a remainder, prior to that which in the general case is R to vanish; because (by hypothesis) the H. C. M. of F. (2) and 1X1

с

is not of less dimensions than the H. C. M. of F. (~) and any term in the first line of (2). The fact, therefore, of R being zero, implies that F. (x) has a common measure with each of the terms in the first line of (2), and that its H. C. M. with any of these terms is of the same dimensions as its H. C. M. with any of the rest. Let L be the H. C. M. of F. (~) and 1X1, L1 that of F. (x) and 2X1, and so on. The terms L, L1, &c., are all of the same dimensions; Li, in fact, being what L becomes on substituting z Y1 for Y1; and so on. Also, since all the factors of the terms in the first line of (2), being factors of Fe+1 (x), are unequal, it follows that all the factors of the terms L, L1 &c., are unequal. This, taken in connection with the fact that F. (x) is a factor of the continued product of the terms in the first line of (2), shows that F. (x) is equal to the continued product of the terms L, L1, &c. Thus the first of equations (9) is established. In the same manner the others can be established. Cor. 4.-Should F. (*) not be equal to a term in (2), an equation of the form,

must subsist; B and A being whole numbers, distinct from zero: and P an expression involving only surds which occur in F. (x), exclusive of Y; while Y2 is what Y becomes when T is changed into zi

2

For, let N be the H. C. M. of F。 (*) and 1X1, N1 that of Fe (∞) and ¿X1, and so on, Q being the H. C. M. of Fe (2) and 1X, Q1 that of Fe (x) and ¿X2, and so on: in which case the terms, N, N1, &c., are respectively what L, L1, &c., (see Cor. 3), become on changing Y into zY; and Q, Q1, &c., are what K, K1, &c., become on changing Y into z Y. Then, in the same way in which equations (9) were found, we can establish the equations.

2Fe (x)=Q × Q1 × Q2 × ...... × Qy-1 •

Now suppose, if possible, that such an equation as (10) cannot subsist. Then, exactly as it was shewn in Cor. 2, [proceeding upon the hypothesis that such an equation as (3) cannot subsist], that any function involving merely such surds as are in F. (x), together with Y1, is in a simple form, we may demonstrate [proceeding upon the hypothesis

that such an equation as (10) cannot subsist] that any function involving merely such surds as are in F. (x), together with Y1 and Y2, is in a simple form. This being premised, we remark, that, in (9), Lis either equal to one of the expressions K, K1, &c., or has a common measure with more than one of them. Let K, be a term in the series K, K1, &c., such that the H. C. M. of L and K. is not of less dimensions than the H. C. M. of L and any other term in the series. Take K, the general form which includes all the terms K, K1, ......" K,, and likewise all the terms Q, Q1, Qs-1; the latter series being derived from the former by changing Y into z Y. Perform the operation of finding the H. C. M. of L. and K', stopping at the point where, in the particular case of L and K., the process comes to an end. If at this stage the remainder be R, and Ry be the corresponding remainder in the case of L and K., the forms of R and

R1 are,

R

R1

=

+ X

λ λι

1

2

+ &c. ;

the expressions being similar to those in Cor. 3. But since R1 = 0, we find (as in Cor. 3) that R 0; it being kept in view that any function which involves merely such surds as occur in F. (*), together with Y2 and Y1, is in a simple form. Hence L has a common measure with every term included under the general symbol K, and therefore it is a factor of 2F. (*) as well as of F. (x): which, since F. (*) and 2F. (*) have no common factors, is impossible. Therefore an equation such as (10) must subsist.

Cor. 5.-The same suppositions being made as in Cor. 4, the following equations must subsist:

and so on; where Y. is what Y becomes when T is changed into z¡T; and P, P1, P2, &c., are expressions which involve only surds, exclusive of Y, occurring in F. (x); and the whole numbers, 1, λ, A1, Ag, &c., are such, that, if λ, a+1 +2, be three consecutive terms in the series, they are related to one another by the equation,

λa+2 = λ λa+1 - βλε

C

a

(12)

For, the first equation in (11) subsists, by Cor. 4. From this we can deduce, by Prop. III. the following, including the first of (11):

and so on; where 'P is the product of an sth root of unity by what P becomes on changing T into 21 T; 'P, the product of an sth root of unity by what P becomes on changing T into zi T; and so on. Raise the first (2c-1) equations in the series (13) to the following powers respectively, viz. : the first to the first power, the second to the Ath power, the third to the Ath power, the fourth to the Ath power; the (c-1)th to the (A)th power, the eth to the (-2)th power, the (c+1)th to the (Ac-3)th power, the (c+2)th to the (B2 Ac-4)th power, the (c+3)th to the (83 c.5)th power, the (2c-2)th to the (p-2)th power, and the (2c-1)th to the (epower. By multiplying together the results thus obtained, we get

.........

Y Y1 Y2

1

Pc-1 Y1 Y2 Y3

Y4

AB-1 ...Y2c-1 ;

where Pe-1 is an expression like P, 1P, &c., involving only such surds, exclusive of Y, as occur in F. (x). But, by (12), we have

and so on; so that the equation obtained above is reduced to

which is the general form that includes all the equations in the

series (11).

Cor. 6.-The (σ + 1)th equation in the series (11) is,

By comparing this with the first of equations (11), we get

But, by Cor. 1, in connection with Prop. III., this is impossible

unless

-

Bo — 1 — ws,

(14)

w being a whole number. Therefore equation (14) must subsist.

PROPOSITION XH.

A given algebraical function of a variable p can always be expressed as an integral function in a simple form; the following conditions being at the same time satisfied: First, that there shall be no surd in the function, of the form,

where T is a chief subordinate of Y, with the index, which is not equal to ; and m is a whole number, not zero, and less than σ; and H is an expression clear of the surd T; secondly, that no two surds, V and V1, principal or subordinate, shall be similarly [see Prop. IX.] involved in the function.

For, should the given function, when rendered integral, be not in a simple form, an equation such as (1) Prop. I. must subsist; all the surds in the equation being surds which occur in the function. Substitute, then, in the function, for Y., wherever it occurs in any of its powers, its value as furnished by (1) Prop. I. Then, when the function is rendered integral, the number of surds present in it, (principal