must subsist; because, in passing from f(p) to oi , V becomes Vı, and M becomes Mı, so that V1 and M, are identical, and hence A1 V1 is equal to B, Mı : and so of the other terms. Therefore from (6), (3), and (4) Pi.. y1 = k Pi . 8 8 Y1 = PROPOSITION IV. If f (P), an integral function of a variable p, be in a simple form, each of its cognate functions is in a simple form. It is self-evident that the Proposition is true for all functions which involve only surds of the first order. Suppose the law to have been found to hold for all functions which do not involve surds above the (n-1)th order : it may then be proved true for a function, f (p), involving surds of the nth, but of no higher, order. For take ø, the general expression which includes all the cognate functions of f (p); one of its particular forms, distinct from f (p), being p1; and suppose, if possible, that $1 is not in a simple form. Then an equation such as (1). Prop. I, di ,12 Y = PY, *Y2... must (Prop. I.) subsist; all the surds involved in the equation being surds which occur in l. We may write this equation in the form, yi P, (1) where yi denotes the continued product of the expressions Y , Y1, &c. Let yı in $i correspond to y in 6, and to Y in f (P) : that is to say, y is what Y becomes in passing from f (p) to $, and yı is what y becomes in passing from 0 to $. In like manner, let Pin di correspond to Q in o, and to R in f (p). Let the the surds Y., Yı, &c., in $i, correspond to Yć, rí, &c., in f(n); and since the surds Y, Yí, &c., have the common index , let their forms be, 1 Y. = V1, Yi Vi, ... Take F (P), a function involving all the surds which occur in the expressions R, Vc, Vi, ...... , Va; and let the particular cognate function of F (p), obtained by making the same changes in the surds involved in F (p) as require to be made in order to pass from f (P) to Ya = V. $i, be Fi (p). Then the surds occurring in F (P) are all of lower orders than Y, Y , &c.; hence they are all of lower orders than the nth. But we are at present reasoning on the hypothesis that the law sought to be established in the Proposition holds for all functions which do not involve surds above the (n-1)th order. Therefore, since F (p), containing only surds which occur in f (p), is in a simple form, it follows that the function Fi(p) also is in a simple form. Now, if we refer to equation (1), we find that the surds involved in P, and all the subordinates of those surds whose powers constitute the factors of yr, occur in the function Fi (p). Therefore, by Prop. III, we can deduce from (1) the equation, Y= kR, k being a constant quantity. But this is an equation such as (1), Prop. I. ; all the surds appearing in the equation being surds which occur in f (p). Such an equation, however, is directly at variance with the hypothesis that f (p) is in a simple form. And hence cannot but be in a simple form. Consequently the law sought to be established in the Proposition holds good for all functions which do not involve surds above the nth order. Since, therefore, the law holds good for functions involving only surds of the first order, and since, on the hypothesis of its holding good for functions involving only surds of orders not higher than the (n-1)', we have shown that it must hold good for functions involving only surds of orders not higher than the nth, it holds good universally. PROPOSITION V. If f (p), an integral function of a variable p, in a simple form, be a root of the algebraical equation, F(x) = 0, in which the coefficients of the powers of x are rational functions of p, then $1, any one of the cognate functions of f (p), is a root of the same equation. For take , the indefinite expression which includes all the cognate functions of f (p); and let F (p), Fif(p)}, F ($1), developed by the ordinary process of involution, and arranged so as to satisfy the conditions of Def. 8, be, F (0) = A + A, Y1 + A2 Y2 + + AY, F (41)= A + ALU + A, U2 + ... ... + Ac UC; where A, A1, &c., are rational ; and each of the terms, Yı, Y2, &c., is either some power of an integral surd, or the continued product of several such powers ; V1, V2, &c., being what Y, Y2, &c., become in passing from 0 to f (p); and UI, U2; &c., what Yi, Y2, &c., become in passing from ¢ to pi. The expression for F{f(p)} can only involve such surds as are present in some of their powers in f (p). And f (p), by hypothesis, is in a simple form. Therefore F{ $(p)}, as exhibited above, is in a simple form. It also satisfies the conditions of Def. 8. But, since f (p) is a root of the equation, F (x) = 0, F { $ (p)} is equal to zero. Therefore, in the expression for F {f(p)}, the coefficients A, A1, &c., must (Cor. 1, Def. 9) vanish separately. Hence, F (+1) = 0); and consequently $1 is a root of the equation, F(x) = 0. Cor.-Let f (p) be an integral function of p, in a simple form ; and let certain surds in f (p), viz. : yi, y2, &c., (in which series of terms, as was noticed in Def. 7, all the subordinates of any surd mentioned are necessarily included), have definite values attached to them; and let the cognate functions of f (p), taken according to the manner described in Def. 7, without reference to the surd character of yi, y2, &c., be 01, 02, 03, Φη. Also let F (2) = 0, be an equation in which the coefficients of the powers of x are rational as far as all surds except yı, ya, &c., are concerned ; that is, the coefficients contain no surds besides yi , 42, &c. Then, if f (p) be a root of the equation, F(x) = 0, any one of the terms, 01, 02, n, (the definite values of yi, y2, &c., being adhered to), is a root of the same equation. For, in this case, in the same manner in which the expressions for F f (p) and F (01) in the Proposition were formed, we get F { $(p)}= A + Aj V1 + A, V2 + &c. F (01) = A + A, U, + A, U2 + &c. ; where A, A1, &c., are rational as far as all surds except yı , Y2, &c., are concerned ; and each of the expressions, Vi, V2, &c., is either some power of a surd in f (P), not contained in the series, yı, y2, &c., or the continued product of several such powers; Ui, U2, &c., being what Vi, V2, &c., become in passing from f (p) to $: the expressions for F { f (P)} and F ($1) satisfying the conditions of > Def. 8. In passing from f (P) to ø, no change is made on A, A1, &c., because the surds entering into these expressions are the same in f (P) as in die But since F{fip)} is equal to zero, the coefficients A, A1, &c., must (Cor. 1, Def. 9) vanish separately. Therefore F (01) = 0); and $i is a root of the equation, F (x) = 0. (To be continued.) ON THE GEOLOGY OF BELLEVILLE AND THE SUR ROUNDING DISTRICT. BY E. J. CHAPMAN, Read before the Canadian Institute, December 17th, 1859. For the information of distant readers, it may be observed that the town of Belleville, in Canada West, is situated at the mouth of the River Moira near the western or closed extremity of the Bay of Quinté. The Trent, a broad and important river, enters this bay at the upper end, about ten miles west of the Moira, or rather constitutes by its exextension, the bay itself. The Salmon River or Shannon on the other hand flows into the same waters some eight or nine miles to the east of Belleville. The observations contained in the present paper apply almost exclusively to the tract of country thus bounded respectively on the east and west by the Salmon River and the Trent; and extending from a short distance along the shore of Prince Edward's County (south of the Bay of Quinté,) to some ten or twelve miles inland or to the north of the Bay. A few remarks, however, furnished by a hasty visit to the back township of Elzevir, are also incorporated in this paper—leaving the geological details of the iron district of Belmont, Madoc, &c., for a future communication. Throughout this tract of country (as indeed almost everywhere within the Province,) the eye is at once struck by evidences of ancient denuding forces of an action both prior and subsequent to the deposition of the Drift ; and, as a corollary to this action, of the much lower level of the land, relative to the water, at a comparatively recent period of geological history. The shores of the Bay of Quinté in very many places, and the high banks or terraces which run, with more or less of interruption, a short distance inland along the course of the abovenamed rivers, and which were evidently washed at one time by waters either salt or fresh, afford abundant proofs of this earlier physical condition of the district. The foundation rock, so to say, of this locality, is the well-known Trenton Limestone. This, although exposed in numerous places, is generally capped by a considerable thickness of Drift clay, sand, and gravel, with boulders of limestone and various gneissoid rocks, such as lie more or less immediately along the northern confines of the tract in question. Around Belleville itself, more particularly, the upper portion of the Drift consists of very finely stratified sand and light-coloured plastic clay, overlying gravel and other coarser materials with boulders of various kinds. The accompanying sketchsection across the River Moira will serve to convey an idea of the extensive denudation to which the Drift has been here subjected. In this section, a is the upper thin-bedded portion of the Trenton limestone, and b and c are the Drift beds. In consequence of this denudation the beds c are only of partial occurence, but I remarked them in several places at considerable distances apart. They are especially well shewn on the side of a hill or steep bank through whicb a street is cut, in the vicinity of the Court-house, Belleville. A deposit of calcareous tufa derived in great part from minute freshwater shells belonging to cyclas, planorbis, and other genera, constitutes a comparatively recent formation extending over a considerable area on the top of the drift bank or high ground on the west side of the river. It marks the site of an old swamp, now drained off. The same modern calcareous formation occurs still more extensively along the foot of the so-called “mountain at Trenton, (where it was kindly pointed out to me by the Rev. Mr. Bleasdell of that village,) and undoubtedly in many other places ; although the above were the only spots in which it came under my personal observation. It may be stated, as a general |