highest order in f (p), it is impossible, after Y has once been disposed of, as above, that it can ever return upon our hands, as it might do, if it were a subordinate of any of the principal surds in C, C1, &c. From the same consideration we selected V, a surd of the highest order in c, C1, &c. We may obviously go on in the manner described, till we have exhausted all the surds that need to be disposed of, in order to make the expression for f (p) altogether an integral function of p. Def. 6. Let f (p) be an algebraical function of a variable p. Instead of Yı, a surd of the lowest order in f (p), having the index write z,Y, in every place where Y, occurs in f (p) in any of its powers, z, being an indefinite mth root of unity. Do in like manner with all the other surds of the lowest order. Again, Y, being a surd of the order next to the lowest in f (p) thus altered, having for its index, and ze being an indefinite nth root of unity, write z,Y, for Y, in every place where Y, occurs in the function in any of its powers. Proceed in this way, till modifications of the kind described have been made upon all the surds in the function, including those of the highest order ; and let the function, after having suffered all these changes, become $ (p). Denote by P1, P2, 08, ......, Ø, the values of $ (p), not necessarily all unequal, that result from taking all the possible values of the indefinite numerical quantities, 21, 22, &c., which have been introduced into the function. These expressions, $1$. &c., may be termed the cognate functions of f (p). As it is important that a clear apprehension be formed of the manner in which we understand the terms Di, 02, &c., we subjoin illustrative examples. Let } 3 / f(p)=(1+1)+(1+ p) Then, 6 (P)=2(1+z. V p) +29(1+2V p). $ :+*( Here there are, including f (p), six cognate functions ; 를 승 1 / 3 'vp = (1 - Vp) + (1-VP) ; 言 2 / 을 to =z' (1-VD)° +z (1-vp)"; where z is a definite third root of unity, distinct from unity. In the three first of these equations, in order that v, pg, and 03, may be definite, we must take a definite value of ✓p, and then also a definite value of (1 + vp)? As a new surd, (1 – vp), occurs in the three last equations, we must fix upon some definite value of this surd, retaining the definite value already assigned to vp; and then 4, 45, and De, will be definitely determined. Had we assumed $ (P)=(P+ vp* –1)* +(p-vpe—I) (p + vp2 – 1), we should have got six cognate functions; but three of them merely a repetition of the other three ; for the three which result from taking vp2 – 1 with the negative sign are the same as those which result from taking it with the positive sign. Def. 7. Suppose that we form the cognate functions of f (P), as described in the previous definition, with this difference, that we now proceed as though certain surds, Y,, Y,, &c., (in such a series all the subordinates of any surd mentioned are necessarily included), were rational. In other words, attach no indefinite numerical multipliers, (as 21, 22, &c.,) to any of the surds, Y,, Y,, &c.; but consider each of these surds as having a single definite value. The cognate functions of f (P), so obtained, may be termed the cognate functions of f (P), taken without reference to the surd character of the surds Y , Y,, &c. For instance, let 1 3 f (P)=(2+p) +(1+VP) +(1+ Vp) + vp; then the cognate functions of f (p), taken without reference to the 11 1 surd character of the surds, VP, P., (2+P ) , are, } } 3 9 11 1/3 23 을 $,=(2+P) +z (1 + vp) +% (1+p) + vp; z being a definite third root of unity. Def. 8. Let f (p) be an integral function of a variable p; and suppose, that, if Y be any surd whatever, principal or subordinate, occuring in the function in its cth power, and having (see Def. 3) the index, c is less than 8. Also, the form of the function being, f(p)=A+A, Y,+A,Y, + ...... +A.Y., where the coefficients A, A, &c., are (see Def. 1) rational, and each of the terms Y,, Y,, &c., is either some power of an integral surd, or the continued product of several such powers, suppose that no two of the terms, Y, Y,, &c., are identical. Finally, if V be any surd, principal or subordinate, occurring in the function in its nth power, and if the form of v be, V =(B+B,Y,+B,Y+..... +BY), where the coefficients, B, B1, &c., are rational, and each of the terms, Y, Y,, &c., is either some power of an integral surd, or the continued product of several such powers, the index of the surd V being !, suppose that no two of the quantities, Y, Y,, &c., are identical. When these conditions are satisfied, the function f (p) may be described as satisfying the conditions of Def. 8. Cor. Any given algebraical function f (p) of a variable p admits of being exhibited so as to satisfy the conditions of the Definition. For should a surd Y, principal or subordinate, with the index , occur in the function in its cth power, c not being less than s, let ws be the greatest multiple of s in c; the excess of c above ws, (which may be zero), being k. Then we may replace Yo by (Y“) Y"; and, since the index of the surd Y is , Y may be written out so as to involve only the subordinate surds of Y. Thus the violation of the first condition of the Definition, involved in the term Y', is got quit of. For instance, Y=(1+vp) = (1 + vp)° + vp (1+ vp)o. Next, should any such quantities as Y , Y,, &c., (see above), be identical, the terms containing the identical quantities, as described, may be combined into a single term. For instance, f(p=p+ {1+p+p+p+vp}'+r{1+pvpapirp} =p+(1+p) {1+(p+78)»p} ? Def. 9. An irrational function, f (p), of a variable p, is said to be in a simple form, when no equation such as, A+BU+CV +......+DY+......+ ET=0,............. (1) can subsist; where the coefficients, B, C, ......, E, all of them distinct from zero, are (see Def. 1) rational ; A likewise being rational; and each of the terms, U, V, ......, T, is either some power of an integral surd occurring in f (P), or the continued product of several such powers; the expression on the left hand side of the equation satisfying the conditions of Def. 8. Let it be observed, that, in this paper, when we speak of a surd occurring in a function, we mean that the surd appears in the function, as a principal or subordinate surd, in some one or more of its powers, but not necessarily in the Thus, the surds which occur in the function, 옥 Nipe – 1), first power. are, vp? 1, and, (P wp3 – 1) The first occurs in its first power; the second, in its second and fifth powers. This being kept in view, we may instance, as violating the condition above mentioned, the function, 3 7 / 출 f (P) = (p + vp? 1) + (P wp2 - 1) ....... ..(2) For the equation, of the form (1), subsists : } Vp? 1) - (P + vp? – 1) (p – vp? – 1) = 0. Hence f (P), as exhibited in (2), is not in a simple form. Cor. 1. The Definition implies, that, should an irrational function of a variable p, in a simple form, and equal to zero, present itself in the form, f (p) = A + BU + CV + + ET, where U, V, &c., are terms of the same kind as in equation (1), and (P + + A, B, &c., are rational, the coefficients A, B, &c., must vanish sepsrately. Also, should f (p) be of the form, f (P) = A + A, V, + A, V, + + A. V. where eacu of the terms, V., V., &c., no two of them ideutical with one another, is either some power of an integral surd, or the continued product of several such powers, while the expressions A, A1, &c., involve only surds distinct from those whose powers constitute the factors of the terms V, V, &c., then [it being understood, as before, that f (p) is in a simple form and equal to zero the cofficients A, A,, &c., must vanish separately. Cor. 2. If f (p), a function of a variable p, be in a simple form, and if A + A, Y, + A, Y,+ ... + A, Y. = B+B, U,+B, U, + ... +B.U ;...(3) where A,, B1, A2, B,, &c., none of them being zero, are rational; A and B also being rational; and each of the expressions, Y,, U,, Y U, &c., is either some power of an integral surd occuring in f (P), or the continued product of several such powers; the expressions, A + A, Y, + &c., B + B, U, + &c., having been arranged so as sererally to satisfy the conditions of Def. 8; then the surd parts, Y, Y., ........., Y. (4) are identical, taken in same order, with the surd parts, U,, U2.........., UL. ;......... (5) and, U, being the part identical with Yı, the rational coefficient B, is equal to the rational coefficient A. What we mean by identical with, as distinguished from equal to, may be shown by an example. The surd vp2 - 1 is equal to the product of the two surds, Vp+1, Vp-1. But the expressions, vp2—1, 1p+1 vp--1, are not identical ; because the only surd which appears in the former is not found in the latter; and the surds which constitute the factors of the latter do not appear in the former. The truth of the Corollary may thus be shown. Should any term in (4), as Yu, be identical with a term in (5), as U,, let the two terms, A,Y, and B,U,, in (3), the latter removed to the left hand side of the equation, be written as one term, Y, (A, -- B.). No other term in (5) can be identical with Y,, for then it would also be identical with U,; but since the expression, B+B.0, +, &c., satisfies the conditions of Def. 8, no two terms in |