where Cj, Cg, &c., are clear of the surds V and T; and H,, Hg, &c., B (9) ...... where i, L, &c., may involve the surd V, but are clear of T; and B none of the terms Li, Lg, &c., are zero; and T , T , &c., are dis λ tinct powers of T. Then, if the form of U be, U-Qi, and it QQ7", when rendered integral, and made to satisfy the conditions of Def. 8, be written t, we have, by (8) and (9), B - + + L,T + L,T + &c, From this equation let the surd to be eliminated, in the same manner in which X, was eliminated from equation (4) Prop. I. The result is, LK + K,L,T + K,L,T + &c. -0. + K" &. B Here, since the surd T does not (except as implicitly involved in 1) appear in t, the expressions K, Kj, &c., are clear of the surd T. Therefore (Cor. 1, Det. 9), the terms LK, K,, K,, &c., vanish separately. But, froin the manner in which K, K,, &c., originated, this implies that L-ct; λ L,T kt 1 L, T = qto; and so on ; C, k, q, &c., being constant quantities. Hence, there cannot be more than one term in the series L1, L2, &c. ; else we should have qL,T - kL,T : $" which (Cor. 1, Def. 9) is impossible. For a similar reason, L must 1 vanish ; and the form of † is, 1 Therefore, if 18 -Òo + r, we have t-LZI", where L, is clear of the surd T ; and r is a whole number, less than but (since s and o are unequal) not zero. But t - QQ,*-Q (U). .: L,T" - Q (U) And, when the expression on the right hand side of this equation is rendered integral, it is clear of the surd T. Therefore, by Cor 1. Def. 9, L, must vanish. Hence t vanishes : which implies that P or A, vanishes. But (by hypothesis) A. does not vanish. Therefore T does not appear in the expression P. In like manner, if, when T, is eliminated from A. (see (1)] by substituting its value as furnished by equation (7), A., made to satisfy the conditions of Def. 8, be written P,, it may be proved, since the equation, B.Y; -- 6m (, Yo) has been shown to subsist, that the surd T does not appear (except as implicitly involved in V) in A.. Ultimately, we get f (p) – P + PU + P, U'+ &c. ; (10) where P, P, &c., are what A, A., &c., in (1), become on substituting for T, its value as furnished by (7); the expression on the right hand side of (10) being clear of both the surds T and T,, except as these surds are implicitly involved in V. Hence the surds T and T, are similarly involved in f (p): which is contrary to hypothesis. Therefore the equation (3) cannot subsist. PROPOSITION XIV. Let the equation, X-0, be an algebraical equation of the fifth degree, in which the coefficients of the powers of u are rational functions of a variable p; X being incapable of being broken into rational factors, that is, factors having the coefficients of the powers of x rational. Then, should the roots of the equation, X - 0, admit of being represented in algebraical functions, they are all contained in the expression, f (p)–A+(A,+B, VČ) (D+D, VC)*+(A,+B, VC)(D+D, VC) 3 +(Az+B3VC)(D+D, VC)*+(A2+BAVC)(D+D, VC);... (1) where C, D, D,, A, A,, B2, A,, B,, &c., are rational functions + + of p. For let f (P), a root of the given equation be reduced (Prop. XII.) to a simple integral form, containing no surd such as Y in equation (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. Take Y, a surd in f (p), not subordinate to any other in the function. Then, if we consider the manner in which the terms of the series, 3-5 (p), F, (2), F, (x), F. («) or X, ... ... ... ... (2) in Prop. VIII, are formed, it appears, that, in an equation of the teha degree, the reciprocal of the index of Y is a measure of t. Hence, in the case before us, the index of Y is }; and from this it follows that the series (2) is reduced to the two terms, X-f (p), X. Besides Y, there can be no surd in f (p), which is not a subordinate of Y; for, if U were a surd in (p), distinct from Y, and not subordinate to any surd in (p), then, since the coefficients of the different powers of x in X are rational, the surd U disappears from X; consequently (Prop. IX.) the index of U is the same with that of Y, nates. *F.(2): : and the surds U and Y are similarly involved in f(p): which is not the case. Hence f (p) contains no surds, except Y and its subordi. Let T be a chief subordinate of Y, with the index . Then, since the coefficients of the different powers of u in X are rational, T disappears from X. Let the factors by the continued product of which X is produced be compare (1) Prop. XI] {x-f (p)} or F.(x), ° F.(2), (3) and, when T is changed into 2, T, 2, being a oth root of unity, distinct from unity, let the terms in (3) be transformed into X, ,X, 8X, Then, since the terms in (1) are the five factors of X, F. (x) must be equal to one of these terins, which (on the principle pointed out in Prop XI.) may be assumed to be ,X,. Consequently (Prop. XI.) Y=PY ; (5) where Y, is what Y becomes when T is changed into z,T; and P is an expression involving only surds in f(p), distinct from Y; 1 being a whole number, neither zero nor unity, less than 5, and such that i = 5w+1, (6) where w is a whole number. Since o is a prime number, the only values of 1, less than 5, which satisfy equation (6), are 1 and 4 And I is not unity. Therefore 1 = 4. Hence o 2; and T is of the form, T- Vi. Next, suppose, if possible, that U is a chief subordinate of Y, distinct from T. By the same process of reasoning as above, it may be shown that the index of U is ; and, if , Y be what Y becomes when U is taken with the negative sign, -U, the equation, Y=Q (,Y*), subsists; where Q is an expression such as P in (5). Therefore PY; -Q (,24). By raising both sides of this equation to the fourth power, keeping in view that the common index of Y, and , Y is }, we get ¡YRY; (7) where R is an expression, like P and Q, clear of the surds Y, and , Y. When U is changed into the negative expression, - U, let the terms in (3) become, 'x, 'X, .... , 'X; (8) and, since F.(2) must be equal to one of the terms in (8), assume (on the principle pointed out in Prop. XI.) that F.(2) – 'x. Take x ", a power of x in F. (2), such that some power of Y is present in its coefficient E; and, E, and E, being the corresponding coefficients of 2" in , X, and 'x, let E, E,, and E,, satisfying the conditions of Def. 8, be, E = B + BY + B.Y + &c., E, = B + B. (, Y') + B. (,Y") + &c. ; where B., B., &c., none of them zero, are clear of the surd Y; B also being clear of Y; and no two terms in the series, Yo, Yo, &c., are identical with one another ; Y, b, be, &c., being what Y, B, BE, &c., become in passing from F.(2) to ,X, ; and ,Y, B, Bc, &c., what these same quantities become in passing from F. (x) to 'x. Then, because , X, and 'X are each equal to F. (2), they are equal to one another. This implies that E, and E, are equal to one another; but (Prop. XIII.) 6.Y; is not equal to B. (, x°). It may be shown, however, exactly as in Prop. II., that the terms, 6Y, b.Y;, &c., taken in some order, are equal to the terms, B. (1 7°), B. (, xo), &c., each to each; and, if the steps of the demonstration be referred to, it will be seen, that, since equation (i) subsists, b.Yę must be equal to to the term B. (17°): which is impossible. Therefore U cannot be a chief subordinate of Y; and T is the only chief subordinate of Y. Again, suppose that U is a chief subordinate of T, with the index ; and, when U is changed into 2,U, 2, being a pet root of unity, distinct from unity, let the terms in (3) become, X, X, (9) the surds Y and T at the same time becoming y and t. Then if E, be the coefficient of 2c in , X, we may put |