(5) are identical. If U, be identical with a term in (4), necessarily distinct from Yı, let that term be Y,; and let the two terms, A, Y, and B, U,, in (3), the latter removed to the left hand side of the equation, be written, Y, (A, — B,). Make all other such modifications as are possible. Then equation (3) becomes, (A - B)+Y,(A, – B)+......... +A Y.-B.U.+ &c.=0);......... (6) where all the terms, Yu, Y., Un, &c., are distinct; so that the expression on the left-hand side of equation (6) satisfies the conditions of Def. 8. Therefore, by Cor. 1, the coefficients, A,-B,,......, A., B. &c., vanish separately. But, since the terms A, B,, &c., are all (by hypothesis) distinct from zero, this shows that there are, in fact, no such terms in (6) as those which we have written, A, Y., — B. U. Hence the terms in (4) are identical, taken in same order, with those in (5). Also, Y, being identical with U,, we have seen that A, is equal to B. PROPOSITION I. If f (P) be an integral function of a variable p, not in a simple form, then an equation, Y λ, ........... (1) must subsist; where Y, Y, &c., are surds, principal or subordinate, occurring in f (p), of the same order, and with a common index ; ,, dg, &c., being whole numbers, less than 8; while P is an expression involving only such surds, occurring in f (p), as are of lower orders than the surds Y., YJ, &c. For, since fp) is not in a simple form, an equation such as (1), Def. 9, Y A + BU + CV + + DY + + ET = 0,...... (2) subsists; all the surds involved in the equation being surds present in f (p). Let Yo = PY,''Y, W...... .X.'. ........ (3) be an equation such as (1), with this difference, that the indices of the surds Y., YJ, &.., are not assumed to be equal to one another; but d, is less than the denominator of the index of Y, de less than the denominator of the index of Yg, and so on. Of the terms, U, V, ......, T, in (2), let those which involve among their factors surds of the highest order in equation (2), be, U, V, ......, Y; and let the sum of A and of those terms, such as ET, in (2), which do not involve surds of the highest order present in (2), be H. Then H + BU + CV +..... + DY= 0. Again, let U= H, X,, V = H, X,,..........., Y = H, X, ; where X, is the continued product of those factors of U, which are powers of surds of the highest order in (2); X,, the continued product of those factors of V, which are powers of surds of the highest order in (2); and so on. Then, putting H + BH, X, + CH, X, + ...... + DH, X, = 0,............(4) let us suppose, if possible, that no such equation as (3) can subsist; and, in connection with this supposition, let us make the hypothesis, that the terms, X, X,, &c., are all distinct from one another. By differentiating (4) with regard to p, we get di Ing (H) } d { log (BH, X,)} H + BH, X, + &c, = 0 (5) dp dp Multiply (4) by the coefficient of BH, X, in (5), and subtract the product from (5). Then hH + h, CH, X, + + h, DH, X, = 0; (6) where the values of h, h,, &c., are H BH,X, BH, X, h, dp and so on. None of the factors of the coefficient of X, in (6) vanish. For C (by hypothesis) is not zero. The equation, H, = 0, is virtually of the form (3), which we have supposed inadmissible. And, if h, were zero, we should have, by integrating the value of hg, BH, X, =kCH, X, X,......... (7) k being a constant quantity, that is, a quantity independent of p. But since X, and X, are not identical, there must be one factor of 1 Xy, as Mo, such that X, either has no power of the surd M as one 1 of its factors, or a power of M^ distinct from the cth Both of these alternatives are included in the assumption that M^ is a factor of X,, r being a whole number, which is not equal to c, but may be zero. Hence, if equation (7) subsist, we have where X is what X, becomes when the factor Mois rejected; and X“ 1 is what X, becomes on the rejection of the factor M^. Since c and r are whole numbers, different from one another, and each less than the prime number 1, we can choose whole numbers, m and n, such that m (c-r) = n1 + 1. Then (BH, X')" mom= (kCH, X")". (9) But this equation will be readily seen, when the expression on its right hand side is rendered (Cor. Def. 5) integral, and made to satisfy the conditions of Def. 8, to be of the inadmissible form (3). Consequently h, cannot be zero ; and therefore the coefficient of X, in (6) is not zero. In like manner it can be shown that the coefficients of all the other terms, such as X,, in (6), are distinct from zero. Again, the coefficients of the terms, H, X,, X., &c., in (6), when rendered integral functions, and made to satisfy the conditions of Def. 8, involve no surd of so high an order as those whose powers constitute the factors of X,, X,, &c. This will be plain if it be considered that the differential coefficient of the logarithm of any power of a surd does not involve, when arranged so as to satisfy the conditions of Def. 8, the surd in question. For instance, ) -P dp 3p(1-P) where the differential coefficient obtained is clear of the surd (1 + V p) ? Since therefore the coefficients of the terms, H, X,, ....... X , in (6), when arranged so as to satisfy the conditions of Def. 8, involve only surds of lower orders than those whose powers constitute the factors of X,, X., &c., and since the coefficients of the terms, X,,........., X , in (6), are all distinct from zero, it follows that equation (6) is of the same character as equation (4). But equation (6) contains one term less than equation (4), X,, having been eliminated. Therefore, in the same way in which equation (6) was derived from (4), we may deduce from (6) another equation of the same character as (6), but with a term fewer. And so on, till ultimately we get bH + 2X=0; where l and b, the former not zero, involve no surds of so high an order as those whose powers constitute the factors of X, But (compare the reasoning by which equation (9) was deduced from (8)] this is virtually an equation of the inadmissible form (3). Hence, in consistency with the hypothesis that equation (3) cannot subsist, it cannot be supposed that the terms, X1, X,, ........., X , in (4), are all distinct from one another. Shonld X, then be identical with X,, let the two terms, BH, X1, DH, XQ, be combined into the single term, X, (BH, + DH.). Make all other such modifications on equation (4) as are possible. Ultimately we get H+X, (BH, + DH, + &c.) + X, (CH, + &c.) + &c.=0...(10) where no two of the terms, X1, X,, &c., are identical. But, by what has been proved, this is impossible, except upon condition that the coefficients of X, X,, &c., vanish separately. Put therefore BH, + DH, + &c. = 0 ...... (11) If we compare this equation with (2), we perceive that it is of the same character as (2), with this difference, that there is no surd in equation (11) of so high an order as some of the surds in equation (2). But, in the same manner in which we derived (11) from (2), we may deduce from (11) another equation bearing the same relation to (11) as (11) bears to (2). And so on, till ultimately one of the equations, such as(10), at which we arrive, contains only one term such as X, with no more than a single term, such as BH,, for its coefficient: from which it follows that B must be zero; whereas all the coefficients, B, C,........., E, in (2), were supposed (see Def. 9) to be distinct from zero. Hence some equation such as (3) must of necessity admit of being formed. Now suppose that the indices of the surds, Y., Y,; &c , in (3), are, 3, 5, &c; and that s, is not equal to s. By raising both sides of equation (3) to the oth power, we may easily [compare the manner in which equation (9) was deduced from (8)] transform (3) into an equation, not involving the surd Y,, B B > Y,= P, Y, Y, ........, Y. where P, is an expression such as P; B, being a whole number less than the denominator of the index of Yg; Bz, a whole number less than the denominator of the index of Y, ; and so on. By continuing this process of reduction as far as necessary, we ultimately arrive at an equation such as (1). Cor. Let each of the terms, Y , Y ,, &c., be either some power of an integral surd, or the continued product of several such powers; while A1, A,, &c., are algebraical expressions, distinct from zero; and A is an algebraical expression not assumed to be distinct from zero. Then, if A + A, Y, + A, Y,+ ...... + A, Y, = 0,......(12) an equation of the form, Y, = PY, (13) must subsist; where P is an expression involving only such surds as are present in the coefficients A, A,, &c., or are subordinates of some of the surds whose powers constitute the factors of Yı, Y,, and Y, is a term in the series, Y,, Y.; m being either unity or zero. For, in the same way in which we eliminated X, from equation (4), we may proceed to eliminate successively the terms Y., from (12). The result of the elimination of Y, is, Y, d A, ...... + A, Y, + &c. = 0...... .(14) dp Here (see remarks in the Proposition) the coefficient of I, when made to satisfy the conditions of Def. 8, involves no surds except such as are found in A, or A,, or are subordinates of the surds whose powers constitute the factors of Y, and Y. Hence the coefficient of Y, in (14) is an expression such as P in (13). Should this coefficient vanish, we have A, Y, = k A, Y,, k being a con &c.; ......... Ig, > { log (A, Y; |