E, = B + B.y + Boy + &c., where y, B, B., &c., are what Y, B, B., &c., become in passing from F. (x) to ,X. Since F. (x) is equal to one of the terms in (9), we may assume (on the principle pointed out in Prop. XI.) that it is equal to ,X; in which case E is equal to Ez ; or, B + B.Y + &c. = 8 + Beye + Boya + &c. ... (10) Therefore (Cor. Prop. I.) an equation of one or other of the forms, Y° =Ly', (11) Y = L, (12) must subsist; where L is an expression involving only such surds as are found in the expressions, B, B, B., B., &c., or are subordinates of Y or y; and y' is a term in the series Yo, Y", &c., yo, yo, &c., distinct from Yo. Let us assume that equation (12) subsists. Then L cannot be clear of the surd t: else all the surds in L would be found in F.(2): in which case, by Def. 9, equation (12) would be impossible. The expression L, therefore, satisfying the conditions of Def. 8, may be written, L = H+H,t, where H and H,, the latter not zero, are clear of the surd t. Hence YS = H + Hit. From this it follows (Cor. Prop. I.) that an equation of one or other of the forms, (13) Y' =hH,t, (14) must subsist; where h is an expression involving only surds which are found in H or H,, or are subordinates of Y or t, and therefore only such surds, exclusive of Y, as occur in F. (x). But equation (13) is impossible, by Cor. 1. Def. 9. Also, should (14) subsist, we should have, (since it is the index of t), Y= ; where h is an expression involving only surds which occur in F. (2). But this is (Cor. 1, Def. 9) impossible. Therefore neither equation (14) nor equation (13) can subsist ; and hence equation (12) cannot subsist. Therefore equation (11) must subsist ; and Y = AH, .... 20 n Boy , 5 we may assume that y' in (11) is a term in the series, yo, y“, &c. ; for, were it such a term as Y", equation (11) would be reduced to the inadmissible form (12). If then we substitute for y in (10) its value derived from (11), we may [keeping in view that no such equation as (12) can subsist] demonstrate, by reasoning similar to that employed in Prop. II. and Cor. Prop. I., that B. Y' is equal to some term, as Be yo, in the series B. yo, &c. Let the fifth powers of B, Yo and B. y. , satisfying the conditions of Def. 8, be, (B. Y° )° = hth (B. y) = k + kit; where h and h, are clear of the surd T; and k and k; are clear of the surd t; and (Cor. Prop. X.) h, is not zero. Therefore, from the equation, h + h, T = k + k, t, T=l, (15) T = (R + R, U + R,U + , R,U' + &c.); + S ; a vel where R, S, R, S, &c., are clear of the surd U. Then, from (15), (v + 0, U + + v Upl) T=(0+0, U + &c.) (S+S, U+&c.)t = (v + V,U + ..... + V, U1) t; where the expression, V + V, U + &c., is generated by the multiplication together of the two expressions, v + 0, U + &c., $ + S, V + &c,; the coefficients, V, V,, &c., being clear of the surd U. Let us P-1 Then equation (15) gives us } 1 Hence, by Prop. X., one of the surds in F. (x), viz. T, is of the form of Y in (1) Prop. XII.: which (by hypothesis) is impossible. Hence U cannot be a chief subordinate of T; and therefore f (p) involves no surds except Y and its subordinate T; the latter being of the form T-C. Consequently f (p) is of the form (1). We may notice a particular form in which f (p) admits of being expressed. By means of equation (5), we can reduce (1) to the following: $(p) = A + (A, + B, VC) (D+ D, vc) 3 } – Cor. 1.—The exact resolution, in algebraical functions, of an equation of the fifth degree, is only possible when X admits of being broken into rational factors, or when the roots can be reduced to the form (1). Hence, in the most general case, the exact resolution, in algebraical functions, of an equation of the fifth degree, cannot in the nature of things be effected. A fortiori, the exact resolution, in algebraical functions, of equations of degrees above the fifth, cannot, in the most general case, be effected. Cor. 2.-In all the cases in which the roots of a quintic equation, whose coefficients are rational functions of a variable P, admit of being represented in algebraical functions, the principles which have been established enable us actually to solve the equation. For, if X can be broken into rational factors, these factors may easily be found ; and thus the solution of the quintic is obtained. Should X not admit of being broken into rational factors, assume f (P) equal to the expression in (1). Substitute for f (P) in X the expression to which x is thus assumed equal ; and let the result of the substitution be, 6+(a, +6, VC) (D+D, v©)*+(,+b, VC) (D+D, v©). +(23 +63 vē) (D+D, VC)* +(+bavē) (D+D, VC)*; where the expressions, b, a, b, ag, b,, &c., as well as C, D, D,, are to be assumed rational. Put 9,=0, 4,=0, ag=0, a=0, b=0, b, =0, 6,=0.bg=0, be=0; and these equations will enable us to find the unknown quantities in the value of f (p): it being taken for granted that the rational roots of an algebraical equation, having the coefficients rational functions of a variable, can always be found. Note. From what has been proved, it appears that the roots of an algebraical equation of a degree higher than the fourth do not, in the most general case, admit of being represented in finite algebraical functions ; and we have seen how an equation of the fifth degree, whose coefficients are rational functions of a variable p, may be actually solved, whenever, in consequence of particular relations among the coefficients, the roots are capable of being algebraically represented. It is easy to extend the conclusions which have been obtained t., equations of every degree ; and, from the principles established in the above Propositions, to show how an algebraical equation of any degree, whose coefficients are functions of a variable P, may be exactly solved, in all cases in which an exact solution in finite al gebraical functions is in the nature of things possible. This we propose to do in a subsequent paper. A POPULAR EXPOSITION OF THE MINERALS AND GEOLOGY OF CANADA. BY E. J. CHAPMAN, PROZR86OR OF MINERALOGY AND GEOLOGY IN UNIVERSITY COLLEGE, TORONTO. PART II. INTRODUCTORY NOTICE, In the first of this series of papers, published in the January Number of the Canadian Journal, we gave a brief review of the more common characters or properties employed in the determination of minerals. The present paper exhibits the practical application of these characters, in the distribution of our Canadian minerals into a small number of easily recognized groups, so arranged as to lead at once to the names of the included substances. By referring to the heads of this arrangement or classification, * as given below, it will be seen that there are four principal groups : A, B, C, and D: the first two containing those minerals which exhibit a metallic aspect ; and the other two containing our glassy, stony, pearly, or earthy-looking minerals. The metallic-looking substances placed in group A are sufficiently hard to scratch windowglass ; whilst those placed in group B, are too soft to effect this. In like manner, the minerals of non-metallic aspect placed in group C, scratch glass ; whilst those placed in group D, are less hard than glass, and are consequently unable to scratch that substance. The term “glass," as employed in this sense, means ordinary windowglass. By these simple characters it is easy to determine in a minute, to which group a substance under examination belongs. This determined, we proceed to a consideration of the sub-groups, 1, 2, 3, &c., of the group in question. In the sub-group or section to which the substance will thus be found to belong, there will probably be some three or four, or perhaps half-a-dozen, other minerals ; but these, it will be seen, are readily distinguishable, one from another, by colour, • The general reader should understand that this classification is a purely artificial one, intended solely to lead to the recognition of minerals by means of their more obvious or easily determined characters-somewhat on the principle of the Linnæun classification of plants. |