where b, b1, &c., are clear of the surd yi. The corresponding coefficient in F2 (x) is 2 Therefore, b1 (≈ — 1) y1 + b2 (z2 — 1) y2 + &c. Since the surds present in this equation are surds occurring in f(p), and ƒ (p) is in a simple form, the coefficients, b1 (z — 1), bą (221), &c., must (Cor. 1. Def. 9) vanish separately. But, since z is an rth root of unity, distinct from unity, r being a prime number, none of the expressions, z Therefore -1, 221, &c., vanish. b1, b2, &c., must all be zero: which is inconsistent with the assumption that the surd y1 is present in the coefficient selected. Therefore F1 (x) is not equal to F2 (x); and we proved that it has no common measure with F2 (x). Therefore no term in (1) is equal to a term in (2); and all the terms, 41, 42. ...., P2n, are unequal. In the same way it appears that all the term3, 1, 2,................, pur, are unequal. The terms, 1, 2,......, Par, thus proved unequal, are the unequal cognate functions of f (p), obtained by giving definite values to the surds in F, [which, from the manner in which F was generated, are necessarily surds occurring in ƒ (p)], and framing the cognate functions without reference to the surd character of these surds. For, in framing the cognate functions, 41, P2,............. the surds in F1 (x), except y1, were considered as definite; and no numerical multipliers (such as Z1, Z2, &c., in Def. 6) were affixed to them. If F contained all the surds in F1 (x), except y1, our point would be easily established. It may happen, however, that ...., nr, all Let t be one of these, Then, in virtue of the s F does not contain all the surds in F1 (x) except y1. Other surds may have disappeared from it, along with y1. if there be such: and let its index be. values that may be given to t, the cognate functions of ƒ (p), taken on a non-recognition of the surd character of those surds alone which appear in F, must include s groups of such terms as P1, P2, .........Pnr, In general, if t, ti, are not in F; and if &c., be the surds in F1 (x), besides y1, which 8 1 &c., be the indices of the surds t, t1, &c., there will be 881 groups of such terms as (3). Still further, without having respect to the surds t, t1, &c., there may be (Cor. 5, Prop. VI.: see more particularly the explanation presently to be given) m distinct groups such as (3): only (as has been proved) the nr functions in (3) are the only unequal terms in all the m groups On the whole, the series of cognate functions of ƒ (p), taken on a nonrecognition of the surd character of those surds alone which are present in F, will embrace m nr 8 81 .... terms, or ms 81 ......... lines of terms such as (3), of which the following may serve as examples: ......... The first of these lines is (3). The second is a cluster of terms, in addition to the nr terms of the first line, obtained without having respect to t, t1, &c., and being a repetition of the values of the terms in the first line; for, in the mnr terms, obtained without reference to t, ti, &c., the unequal terms which constitute the series (3) are all repeated (Cor. 5. Prop. VI.) the same number of times. The third line of (4) contains the terms in the first line, transformed by changing t into z1t; z1 being an sth root of unity, distinct from unity. And those in the last line contain the terms of the second line, transformed by a similar change of t into z1 t. Now it can be shown that the terms of the third line are equal, in some order, to those of the first, each to each. For, since t, present in F1 (x), disappears from F, it follows that the continued product of the factors of F, viz. : F1 (x), F2 (x), &c., remains the same when it is substituted for t. That is, the factors, are the same, taken in some order, with the factors, Hence the terms in the third line of (4) are, in some order, equal to those in the first line, each to each. In the same way it may be proved that all the mnrss, ..... cognate functions above described, are merely repetitions of the values of the functions in (3). Hence the terms in (3) are all the unequal cognate functions of ƒ (p), obtain ......... ed by giving definite values to the surds in F, and taking the cognate functions without reference to these surds. Take u1, PROPOSITION VIII Let an equation of the mth degree, whose coefficients are rational functions of a variable p, be, X = 0; X having no rational factors; and let an algebraical root of this equation, in a simple integral form, arranged also so as to satisfy the conditions of Def. 8, be f(p). Take u1, a surd in ƒ (p), not a subordinate of any other surd in the function, with the index; and let the cognate functions of ƒ (p), obtained by successively changing u1, wherever it occurs in f(p) in any of its powers, into w1, z1 u1, z u1, z1 being an nth root of unity, distinct from unity, be, n-1 Let F1 (x) denote the continued product of the terms, x — 41, x —42, -Pn. The coefficients of the various powers of x in F1 (x), made to satisfy the conditions of Def. 8, are (Cor. 3, Prop. VI.) clear of the surd u; and the terms, P1, P2,... Фи constitute (Prop. VII.) the series of the unequal cognate functions of ƒ (p), obtained by affixing definite values to all the surds in F1 (x), [which are necessarily surds in ƒ (p)], and taking the cognate functions without reference to the surd character of the surds so made definite. Should F1 (x), which is clear of the surd u1, not have the coefficients of the powers of a rational, let u2, a surd in F1 (), not a subordinate of any other surd in F1 (x), with the indev, be successively replaced by us, Zq Uq, Z2 U2,..., Zq u2; z2 being an rth root of unity, distinct from unity; and, in consequence of these alterations, let F1 (x) become successively F1 (x), “F1 (x), ̊F1 (x),........., F1 (x); the functions which are 1, 2,........., Pn, in F1 (x), becoming 2 r-1 2 3 on. T P " n + 1 n+g' &c., in F1 (x); and so Denote the continued product of the terms, F1 (x), 2F1 (x),................. F1 (x), when the result is made to satisfy the conditions of Def. 8, by F2 (x), which is (Cor. 3. Prop. VI.) an expression clear of the surd ug, and such (Prop. VII.) that the functions, P1, P2,..., [the nr ar factors of F2 (x) being x-1, X-P2, ......, X — ], constitute the nr series of the unequal cognate functions of ƒ (p), obtained by assigning definite values to all the surds which are found in F2 (x), [these being surds of necessity present in ƒ (p)], and taking the cognate functions without reference to the surd character of these surds. In the same manner in which F1 (x) was derived from ƒ (p), and then F2 (x) from F1 (x), derive Fs (x) from F2 (x), and F, (x) from F3 (x), and so on, till an expressión Fa (x) is reached, in which the coefficients of the powers of x are rational. The expression Fa (x) shall be identical with X. For, if the factors of Fa (x) be, x-1, x— фа M then, since the coefficients of the powers in ≈ in Fa (x) are rational, the functions P1, P2,... Φ constitute (Prop. VII.) the entire series. of the unequal cognate functions off (p). But the entire series of the unequal cognate functions of f (p) is identical (Cor. 6. Prop. VI.) with the series of the roots of the equation, X = 0. Therefore F(x) and X are identical. let the factors by whose continued product Fe+1(x) is generated, be, 3 Fe (x), 'Fe (x), 3Fe (x), ...... 'Fe (x); (2) 3 с where 'Fe (x), Fe (x), &c., are what Fe (2) becomes, on substituting successively for Y, a surd in Fc (2), not subordinate to any other surd in the function, and having the index, the values z Y, zY, &c.; z being an 8th root of unity, distinct from unity. Let U be a surd in Fe (x), distinct from Y, and not subordinate to any other surd in Fe(x), with the index. Then if the surd U disappear from c+1 σ F (x), σ is equal to s, and the surds Y and U are (as we may express it) similarly involved in the function Fe (x): by which we mean, that, when the function is arranged according to Def. 8, whenever one of them appears in the function in any of its powers, it occurs mul tiplied by a power of the other; as, Y by U11, Y3 by U31, 'Y3 by $1 U1, and so on; the pairs of equations, hλ=ks + ß, κλι = 98 + β1, Hλ1 = Q 8 + 81, (3) (4) and so on, subsisting; where h, H, k, K, q, Q, &c., are whole numbers, each less than s To prevent misunderstanding, we may instance the function, ƒ(p) = p2 (2 + p)} + {6+p3 (2+p)3 } { 2 + [p + p2 (2 + p) 1 as one in which the two surds p7 and (2 + p)7 are similarly involved. For, calling the former Y and the latter U, we have s=σ=7, λ=5, A1 = 1, B3, B1 = 2, 81, and d1 = 3. Consequently equations (3) and (4) become, λι where integral values of h, H, &c., less than 7, can be found: We proceed with the proof of the Proposition Let , be a σth root of unity, distinct from unity; and when U is changed into zU let the terms in (2) become, 2 , *f.(x) .... (5) Since U disappears from Fe+1 (x), the continued product of the terms in (2) is not affected when we replace U by 21 U. Therefore 2Fc a 8 Fc (2) × Fe (x) × ... × *Fc (x) = ƒe (x) × ƒe (x) × ... × ƒ。 (x). Hence, either Fe (x) is equal to one of the terms in (5), or it has with one of them, as fc (x), a common measure, of less dimensions, as respects x, than Fc (x). Suppose, if possible, that F. (2) is not equal to any term in (5); and that L is its H. C. M. with fe (2). The ex a |