a E th pression L involves only such surds as occur in Fc() or fc (2); but, k (a-1) YU where such terms as bı are clear of the surds Y and U, and not zero; and no two terms such as that written Yk U are identical; k not being zero. Since 8 is a prime number, and [Fc (2) satisfying the conditions of Def. 8] k is less than 8, we can find whole numbers, w and wi, less than s, and such that wik—w8+1 :: Y-(Y°)""Y","; or, if y be represented by V, Y - (Y*)v" 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 F. (1), introducing in its room powers of V, but no powers of any other surd that was not previously in the function. Let F. (2), as thus exhibited, be written Fi(2). A term in Fc (2) is 6, V U'. Should i + k 1 Z U' + &c., 1 may now be neither zero nor unity, write V, for U'; and, as above, all the 1 k (a-1) 2 k (8-1) 1 %1 (6) If I were zero, this would make zk(2-1) equal to unity : which, since the numbers, k, a-1, are less than s, and z is an opth root of unity, distinct from unity, is impossible. Therefore 1 is not zero ; and hence F. (2) can be exhibited in the form F. (x). Equate the coefficients of 2 in F. (20) and fc (); this latter expression being what fe () becomes when Y and U are eliminated, and V and V, are introduced in their room. Then the equation (6) still holds. But such an equation implies, that, 2 being an eth root of unity, and zı being a wh root of unity, distinct from unity, o=8. Again, suppose V to occur in its h power in any term of F. (w), so that b2 V Vi is a term in the coefficient of some power of r. Then, by reasoning as above, we get +6 VU 2, + &c. 0 1 th 1-21 kh-ws glh-w18 where ws Hence, by a comparison of (6) and (7), 1 (hh) = 0 :: h=hı :: V-V. Hence bą v"v" becomes by V" vi: which, again, by returning from V and Vito Y and U, becomes 6 Y U is the greatest multiple of s in kh, and wis is the greatest multiple of sin lh ; 6 being an expression clear of the surds Y and U. consequently Fc (w) may be written, Fc () = {BY (8) 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 Ulh-w, , included under the symbol 2. But equation (8) implies that the surds Y and U are similarly involved in Fc (2). kh-W8 ; kh-ws PROPOSITION X. Let f (p) be an integral function of a variable p, in a simple form, satisfying the conditions of Def. 8; end let AY^ – BU^; ........... (1) where Y is a surd in f (p), with the index s; and A is an expression, not zero, involving only surds, distinct from Y, which occur in f(P); 1 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, 2 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, 1 Y = (HT" .(2) where H is an expression clear of the surd T; and m is a whole number, less than o. Also, o is not equal to 8. For, let o 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 Yo, except T and Y; and let $', arranged so as to satisfy the conditions of Def. 8, be, Vol. V. M 2 2 21 '-D D + D, 2, T + D2 zi T + + Do.12-27-1 where is an indefinite oth root of unity; and D, Di , &c., are clear of the surd T. Then (AY")* – D+DT + D2 ro + &c ; - + &c. 0. This equation involves only such surds as occur in f (p). Therefore (Cor. 1, Def. 9) the coefficients, D. (1 – 2), D, (1 – 2), &c., vanish separately. But, since o is a prime number, and z is a oth root of unity, distinct from unity, none of the terms, 1 - 2,1 – 2, &c., vanish. Therefore Di , Dą, &c., must all vanish ; and 1 A Y` = D. Raise both sides of this equation to the oth power ; r and n being whole numbers such that g = ns + 1. Then, (A Y^)" = (- (AYY-(0) 1 or, P Y + bo.1 Tol Q" ; (3) 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. Let the forms of P and Y be, P= 6 + 6, T + b2 T + (4) Y = (h+ hn T + he + ..... + horo. (5) where b, bu, &c., h, hi , &c., are clear of the surd T. Suppose, if possible, that the terms bu, b2, &c., are all zero. Then P=b; and, 6h + 6 h, T + 5 h, TP + &c. Q. But since Q is clear of the surd T, the coefficients, 6 hq, 6 ha, &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 Therefore all the terms, hi, hq, &c., vanish : which (since T is a subordinate of Y) is impossible. Hence at least one of the terms, b, b2, &c., as bc , does not vanish. But this leads to the conclusion that all the terms, b, bı , bz, zero. (6) except be, must vanish. For, from (3) and (4), we have, Y b + be YT + 6, Y T QS. .و ہوا 1 + + &c. From this equation eliminate the surd Q3, in the same way in which X, was eliminated from equation (4) Prop. I. The result is, + bc Y To E + bp YT' E, + &c. 0. The conditions necessary in order that E and E, may both vanish, are, k Q! bp YT" - QUBE k and q being constant quantities; and these equations give us, q be To – kb.T": which, To and T' being distinct powers of T, not exceeding the (0-1)th, is (Cor. 1, Def. 9) impossible. Therefore be is the only term in (6) which does not vanish ; and, from (3), (4), and (5), csta cs+m + &c. C8 + e bh T Q ...... (7) If hm be a term in the series, h, hi , &c., which is not zero, all the other terms in that series vanish. For, if he be another term, let = wo + ß, and, cs + m = W10 + 8; where ß and 8 are whole numbers, less than o. Then, since e and m are not equal, and each of them is less than o, ß and 8 are not equal. And so likewise as regards the other terms. Therefore (Cor. 1, Def. 9), all the coefficients, b, ha, bm, &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, hi , &c., except hm, or (as we may write it) H, must vanish ; and Y is reduced to the form, |