must be heated in a test-tube over a small spirit lamp, and after decrepitation has taken place, one of the resulting fragments may be exposed to the blow-pipe flame as already explained. (b) The test-fragment may change colour (with or without fusing) and become attractable by a magnet. Example, carbonate of iron. This becomes first red, then black, and attracts the magnet, but does not fuse. Iron pyrites on the other hand becomes black and magnetic, but fuses also. (c) The test-fragment may colour the flame. Thus, most copper compounds impart a rich green colour to the flame; compounds containing baryta, and many phosphates and borates, with the mineral molybdenite, colour the flame pale green; sulphur, selenium, lead, and chloride of copper colour the flame blue of different degrees of intensity; compounds containing strontia and lithia impart a crimson colour to the flame; some lime compounds impart to it a paler red colour; soda compounds, a deep yellow colour; and potash compounds, a violet tint. (d) The test-fragment may become caustic. Example, carbonate of lime. The carbonic acid is burned off, and caustic lime remains. This restores the blue colour of reddened litmus paper. It also imparts if moistened, a burning sensation to the back of the hand or other sensitive part. (e) The test-fragment may take fire and burn. Example, native sulphur; common bituminous coal, &c. (f) The test-fragment may "volatilize," or dissipate in fumes, either wholly or partially, and with or without an accompanying odor. Thus, grey antimony ore volatilizes with dense white fumes; arsenical pyrites volatilizes in part, with a strong odor of garlic; common iron pyrites yields an odour of brimstone, and so forth. (g) The test-fragment may fuse, either wholly, or only at the point and edges; and the fusion may take place quietly, or with bubbling, and with or without a previous "intumescence" or expansion of the fragment. Most of the so-called zeolites, for example, (minerals abundant in Trap rocks), swell or curl up on exposure to the blowpipe, and then fuse quietly. Lepidolite fuses with great bubbling, and colours the flame red. Feldspar only melts on the edges, at least, in ordinary cases. (h) The test-fragment may remain unchanged. Example, Quartz, and various other infusible minerals. The Water-test.-Many solid minerals contain a considerable amount of water, or the elements of water, in some unknown physical condition. Gypsum, for example, contains 20.93 per cent. of water. In order to ascertain if a substance yield water, we chip off a fragment (of about the size of a small pea) and heat this in a common test-tube (or better, in a small "bulb-tube" or glass tube closed and expanded at one end, as shown in the accompanying figure) over the flame of a little spirit lamp. If water be present, it will rise and condense in the form of a thin film, or in small drops, on the cold neck or upper part of the tube. When the moisture begins to appear, the tube must be held in a more or less horizontal position, otherwise a fracture may be occasioned by the water flowing down and coming in contact with the hot glass. A small FIG. 22. spirit lamp may be made by fitting a piece of glass tubing an inch long (to serve as a wick holder) into the cork of any short, stout bottle. A proper lamp, however, with a glass cap to prevent the evaporation of the spirit when the lamp is not in use, can be purchased for a quarter of a dollar. This concludes our review of the more common characters possessed or exhibited by mineral bodies. The application of these characters to the actual determination of Canadian minerals, by means of an original Tabular Distribution or Arrangement, will be shewn in the next number of the Journal. ERRATA. Figure 7 (on page 7) has been accidentally printed in a reversed position. of arsenic." Page 15, line 5 from bottom, for "which," read "whilst this.” RESOLUTION OF ALGEBRAICAL EQUATIONS. Proof of the impossibility of representing in finite algebraical functions, in the most general case, the roots of algebraical equations of degrees higher than the fourth; with methods for finding the roots of equations of the 5th, 6th, 7th, &c., degrees, in those cases where the coefficients in the given equations involve a general or variable quantity, but where, in consequence of relations subsisting between the coefficients, the roots of the equations happen to admit of being exactly represented in finite algebraical functions. BY THE REV. GEORGE PAXTON YOUNG, M.A., Read before the Canadian Institute, 19th February, 1859. DEFINITIONS. Def. 1. In the functions which are to be considered, a variable is involved; and, when quantities are spoken of as rational or irrational, the meaning always is, rational or irrational with respect to the variable. Thus, c being constant, and p variable, the former of the expressions, c√P, √c+p, is surd or irrational; and the latter, rational. Def. 2. Surds may be distinguished as of different orders. The nth root of a rational expression, n being a prime number, distinct from unity, is a surd of the first order. But the nth root of a rational expression, when n=nnn...n, each of the numbers, n1, ng, &c., being a prime number distinct from unity, is a surd of the sth order. Again, the nth root of an expression involving surds of the sth order, but of no higher order, when n=n1nn ̧.....n1, each of the numbers, n1n, &c., being a prime number distinct from unity, is a surd of the (8+)th order, and so on. Thus, the first of the expressions, 2 (c+p)*; {(c+p}* +p}"; o*+ [{(c+p}) * +p* } * + √p]3 3, is a surd of the third order; the second, of the fourth order; and the third, of the seventh order. Def. 3. Every surd of a certain order is formed by the extraction of some root, (as the nth), of an expression involving only surds of the order immediately inferior, n being a prime number, When we 1 n speak of the index of the surd so formed, the fraction is meant. 1/5 For instance, if we regard (c+p) as generated by the extraction of 1} the fifth root of (c+p), it is a surd of the second order, with the index. From another point of view, it might be described as a surd of the second order, with the index }. Def. 4. In the case of a surd of a certain order, we may distinguish the principal surd from its subordinates. Thus, under the principal surd, (c+ √p), is involved the subordinate p. Under the principal surd, 2 [{o+ (1+p3 * } * +(2+p)3] 3, e being a constant quantity, are involved the subordinates, {c+ (1+p") *}*, (1+p")', (2+p)3, 2 } + the first appearing in the principal surd only in its fifth power; and the second only in its second power. A surd which is a subordinate of the surd Y, but is not a subordinate of any surd which is itself subordinate to Y, may be termed a chief subordinate of Y; while those surds which are subordinates of surds subordinate to Y may be called secondary subordinates of Y. Def. 5. An integral function of a variable is one in which no surd, principal or subordinate, occurs as the denominator, or a term in the denominator, of a fraction. For instance, c being constant, and p variable, the first of the expressions, ; is an integral function of p; but the two last are not. Cor. A given algebraical function f(p) of a variable p always admits of being exhibited as an integral function. For, reduce the function to the form ; where each of the quantities N and D is the sum of a rational expression, which may be zero, and of a finite series of terms, each of them the product of a rational coefficient by some power of an integral surd, or by the continued product of several such powers. Take Y, one of the surds of highest order present in any of its powers in the function; and arrange the terms in N and D according to the powers of Y not exceeding the (m-1), being the index of the surd Y. Then 2 ƒ (p)= a+a,Y+a2Y +...+am-1 I 2 b + b1Y+b2 Y2 +...+6m-1 Y 2 sidered as contained in the term, a, Y. Assume m+2 where the coefficients, b, a, b, a,, &c., may involve powers of any surd inf (p), except Y. No powers of Y higher than the (m.1)th are written; because, for instance, if there were a term AYTM+2 in the numerator, A being an expression clear of the said Y, it might be written, (AY) Y. But Y" may be written so as to involve only the subordinate surds of Y; and hence the term AY may be con m m m +2 a+a,Y+ &c. m-1; m-l ·=c+c1Y+......+cm.1Y ; and, when the expressions, b+b,Y+ &c., c+c, Y + &c., are multiplied by one another, let the product, arranged according to the powers of Y not exceeding the (m-1), be, d+dY+ &c.; where d, d1, &c., are clear of the surd Y. Then d=a, da......, dm-1am-1. = Then the function may be written, a+a,Y+&c=d+d2Y+......+dm-1Y. Determine them unknown quantities, c, c,...., Cm.1, by the m simple equations, 2 n-1 h+h ̧V + h2V2 + ...... +h2-VTMˆ'; n-l m-1 ƒ (p)=c+c,Y+c2Y® +&c.; where the coefficients, c, c,, &c., are clear of the surd Y. Again, let a surd of the highest order present in any of its powers in the coefficients c, c1, &c., be V; and its index. By the process already exemplified, we may find, for each of the coefficients, c, c1, &c., an equivalent expression such as n where h, h1, &c., are clear of the surds V and Y. Let it be remarked, that, in consequence of our having commenced with Y, a surd of the |