Immediately connected with the serpentines, there sometimes occurs a white compact rock, remarkable by its great hardness, and a density of 3.3 to 3.5. Analysis shows this rock to be a pure limealumina garnet, in some cases, however, mingled with another silicate which appears to belong to the amphiboles. This garnet is sometimes blended with serpentine, and at others forms distinct beds. In its general aspect it resembles closely the saussurite of the associated euphotides, and has probably often been confounded with that mineral by previous observers. Hence the densities of 3-2 and 3.3 assigned by different mineralogists to the saussurites of the Alps, while Delesse has shown that the true saussurite of the euphotide of Mount Genèvre, like that of the Vosges, is a felspar.

The magnesites of this region form great beds; they are crystalline, and consist of carbonate of magnesia with some carbonate of iron, and contain as imbedded minerals in some cases grains of quartz, in others felspar and talc, and at other times serpentine, but always holding chrome and nickel, the latter as a greenish carbonate, in the joints of the rock, or in the form of nickeliferous pyrites.

These magnesian rocks are not confined to the altered portions of this formation; beds of siliceous dolomite holding protocarbonate of iron are found, interstratified with pure fossiliferous limestones, near Quebec. The reaction between silica and the carbonates of lime, magnesia, and iron, which takes place at no very elevated temperature, in the presence of water, producing silicates of these bases with evolution of carbonic acid, enables us to understand the process which has given rise to the pyroxenes, serpentines, and tales of this formation, while the argillaceous limestones, which are not wanting, contain all the elements of the garnet-rock.

The general conclusion deduced from these inquiries, and sustained by a great number of analyses, which I hope soon to submit to the Society, is, that the metamorphism of these Silurian strata has resulted from the chemical reaction, in the presence of water, of the elements existing in the original sedimentary deposits.

V. "On Determinants, better called Eliminants." By Pro

fessor FRANCIS NEWMAN, M.A.

Communicated by Dr.

BOOTH, F.R.S. &c. Received March 6, 1857.

(Abstract.)

1. This paper aimed at recommending the introduction into elementary treatises of the doctrine of Determinants; which, following Professor Boole, it called Eliminants. It exemplified the great aid to the memory which the notation affords. It undertook to show, that if only so much of new notation be used, as is needed in elementary applications, the subject becomes full as easy as the second part of algebra. The method of proceeding recommended may be understood by the following concise statement.

If n linear eqq. are given, connecting n unknown quantities; and every eq. is represented by A+B‚μ‚1⁄2 +C‚μ≈3+ · · · +N=P, (where r is 1, 2, 3 ... n in the several eqq.), then, solving for any one of the unknowns, we of course obtain a result of the form ma=a. Very simple considerations then show, that m and a will be integer functions of the coefficients: namely, it is easy to prove, that if this is true for one number n, it must needs be true also for the number (n+1); and consequently is generally true. Next, the same analysis exhibits, that m=0, is the result obtained, when PP, P,... P, all vanish moreover, that if the system presented for solution be the (n-1) eqq.

Bv + B+ B+...+B=0

Nv+N2v2+N3v3+...+N2=0.

and the solutions are denoted by

m'v=a1; m'v2=aq; ... m'vn-1=an-1;

we get the relations

m=Aa1+Aga2+ A ̧αz +...+Aμ-1a2-1+A„m';'

a=Pa1 + P2a2+P3@z + ... + Pn−1an-1+P„m' ;

out of which flow all the rules for the genesis of Eliminants, and the application of them to solve linear eqq. of any degree.

In adapting the theory to the proof of elementary propositions, as, in forming the Product of two Eliminants, the paper urged the utility of the principle, that every Eliminant is a linear function of any one of its columns, and also, of any one of its rows;—which principle may often be so applied as to show by inspection, à priori, that certain constituents are excluded from this and that function, and thus enable us to obtain its value by assuming arbitrary values for such constituents. It deprecated (at least for elementary uses) the notations used by Mr. Spottiswoode* and others, not only as involving needless novelty to learners, but because no page can be broad enough to afford to write

(1, 2)(1, 1)'+(2, 2)(1, 2)'+(3, 2)(1, 3)' instead of BX+bY+ßZ, and because accents, so related, are hard to see in a full page, and the general aspect of every element is so like that of every other element, that the fatigue of reading soon becomes confusing and intolerable.

2. But the main topic of the paper was to advocate the use of Eliminants in Geometry of three dimensions, especially in every systematic treatise on Surfaces of the Second Degree. Various illustrations and results were given, which the writer believed to be new ; on which account, some of them may be briefly noticed here.

Problem. “To find the length of a perpendicular p, dropt from a given point (a b c), on to a given plane lx+my+nz+p=0; when the axes are oblique, and the cosines of the angles (xy)(xz)(yz) are given; viz. D, E, F.”

Result. Take G and H to represent the eliminants

When p is given, this eq. determines the relations between 1 m np, which are the test, that the plane may touch a sphere given in position.

* It may be right to state, that Mr. Newnan opened the paper by a grateful and honourable recognition of Mr. Spottiswoode's labours.

VOL. VIII.

2 K

Problem. To analyse the forms assumed by the locus of the

general eq.

Ax2+By+C+2Ax+2By+2C

+2Dxy+2Ex: +2Fyz+G=0 (axes oblique).

the common treatises (only without this notation) it is shown that when V is finite, the surface (if real) has a centre. It is here added, that when W is negative, the curvature is everywhere towards the same side of the tangent plane; when W vanishes, the tangent plane coincides with the surface in one straight line; but when W is positive, the surface is cut by the tangent plane in two intersecting straight lines, and the curvature bends partly towards one side of the tangent plane, partly towards the other.

Hence it appears that we have different sorts of surfaces, by combining V=0 or V=finite, with W=0 or W=positive, or W=nega

tive.

The locus is imaginary, if W is >0, A and B finite, CG—C22>0,

The locus is degenerate, if of ABC one at least (as C) be finite,

vanish, and if at the same time D=0, and E : F: C2=2A, 2B, : G. Problem. To investigate the nature of the plane intersections of the surface.

Result. If the cutting plane be la+my+nz+p=0, the section is

a hyperbola, parabola or ellipse, according as

tive, zero, or negative.

ADE!
DBF m
EFC n

is posi

I m n o

In a non-centric surface, where V=0, we readily find that the former of these eliminants has the same sign as (D2—AB); and consequently, that non-centric surfaces cannot have sections of opposite species. It also appears, that to determine in a non-centric surface the parabolic sections, we must take l m n such as to verify one of the three eqq.

Problem. To determine the circular sections, when they exist. Result. Take the larger question, of ascertaining when two surfaces of the second degree intersect in a plane curve. Denote the coefficients of the second surface by accents. Put a Ap-A'; B=Bp-B'; y=Cp-C'; &c. and determine p by the eq.

=

Then l m n will be determined (when the surds are real) by the proportion

1: : m : n= √ (e2—ay)+e: √(p2—ẞy) +p: y.

To apply this to the problem of circular sections, it is only necessary to suppose the second surface to be a sphere.

The surface becomes one of Revolution, if (with oblique axes) either system of three eqq. is fulfilled:

If out of each triplet we eliminate p2 and p, (for it seems easiest to treat these as independent variables,) the result is two eqq. (expressible by eliminants), which are the two general conditions for a surface of revolution.