In a non-centric surface, where V=0, we readily find that the former of these eliminants has the same sign as (D3— 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

l : m : n= √ (e2 —ay)+e: √(p2—ẞy)+¢ : 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.

Problem. To find the system of rectangular conjugates. This of course is cardinal, and is treated everywhere: but is made far easier by Eliminants, as follows. Let us inquire after that diameter, common to two given concentric surfaces, which shall have its conjugate planes the same for both.

Take the centre for the origin, and x=mz, y=nz for the common diameter sought. Then the central planes conjugate to it in the two surfaces are

or

(Am+Dn+E)x+(Dm+Bn+F)y+(Em+Fn+C)≈=0

(A'm+D'n + E')x + (D'm+B'n+F')y +(E'm+F'n+C')z=0. ] To identify these two planes, let

Am+ Dn+E Dm+Bn + F Em+Fn+C 1

A'm+D'n+E

=

==

D'm+B'n+FE'm+F'n+C'

am tôn ta ôm+Bn+=em+on+y=0.

€=

Eliminate m, n, and you find that p is to be determined by the very

same eq. as in the preceding; and since its eq. is of the third degree, it has always one real value.

Next, let the second surface be a sphere, and you find at least one diameter of the first surface perpendicular to its conjugate plane. Make this diameter the axis of x, and take for the axes of y and z the two principal diameters of the section in the conjugate plane. Then D=0, E-=0, F=0; so that the general eq. is reduced to Ax2+By+Cz2+G=0. Moreover, the system of axes is now rectangular hence the axis of y, and that of z, equally with that of æ, are each perpendicular to its conjugate plane, and the eq. for p must have three real roots, corresponding to these three axes.

:

We might similarly investigate "the conditions of contact for two concentric surfaces;" which, when one of them is a sphere, gives the cubic whose roots are a2, b2, c2, principal axes of an Ellipsoid.

Problem. To discuss the results of Tangential Co-ordinates. [This expression is employed as by Dr. James Booth in an original tract on the subject.]

Put

P=Ax+Dy+E≈ + Ag

Q=Dx+By+ F≈+B2

R=Ex+Fy+Cz+C2
S=Ax+B2y+Cq≈ + G

Then Pr+Qy+Rz+S=0 is the eq. to the surface, and Px'+Qy' +R+S=0 is the eq. to the tangent plane at (az). Hence if x'y'z' are the three tangential co-ordinates (or intercepts cut from the co-ordinate axes by the tangent plane) we have Pa'+S=0, Qy'+S=0, R+S=0. Let nz be the reciprocals of x'y' z'. Then P+S=0, Q+nS=0, R+¿S=0; and the eq. to the surface becomes Ea+ny +-1=0. Restore for PQR their equivalents; then eliminating xy S you get

A D E A E ૐ
D B F B2 η

general eq. to the surface, with axes oblique.
If the last eq. (developed) be represented by

a2+bn2 +c52+2a25+2b2n+2c25+2dEn+2e85 +2ƒn<+g=0, it is not difficult to obtain a system of eqq. in which abc...Eŋ play the same part, as just before did ABC...xyz. Whence again we have,

a d e Az x
db fb Y

which is the original eq. of the surface under the form of an Eliminant. The most arduous problems (as Dr. James Booth has shown) are often facilitated by these co-ordinates; but without Eliminants, the eqq. cannot be treated generally and simply.

The paper likewise contained the application of Eliminants to tan

gential co-ordinates in Curves of the Second Degree; and urged that eliminants ought to be introduced into the general treatment of these curves also, if only in order to accustom the learner to their use and gain uniformity of method. Thus, if the general eq. be

Ax2+By2+C+2Ex+2Fy+G=0,

then V=0 is the test of degeneracy.

March 26.-Major-General Sabine, R.A., Tr. and V.P., in the Chair. The following communication was read:

"On the Theory of the Gyroscope." By the Rev. William Cook, M.A.

The explanation of the movements of the Gyroscope (as well as its mathematical theory) is founded on the principle enunciated in the two following verbal formulæ.

f towards
from

}

I. When a particle is made to move a plane by any applied force, but in consequence of its connexion with some rigid body on the same side of the plane, loses some of its momentum in a direction perpendicular to the plane; all the momentum so lost is imparted to the rigid body, which is consequently impelled towards the plane.

{

from

II. When a particle is made to move

towards a plane by any }

{from

applied force, but in consequence of its connexion with some rigid body on the same side of the plane, receives an extra momentum in a direction perpendicular to the plane; all the momentum so gained is taken from the rigid body, which is consequently impelled [from

towards

the plane.

The mass of the disc of the gyroscope is supposed to be compressed uniformly into the circumference of a circle of given radius (r), and to revolve round an axis with a given uniform angular velocity (w). To facilitate the arithmetical computation of the formule, masses are represented by weights; so that any effective accelerating force f is supposed to be due to a pressure P acting on a mass W, and their relation expressed thus, f=

Pg

W

cro

being the

The mass of any arc of the circle is denoted by; angle at the centre, and c the mass of a given length of the circumference. The terms of all the formulæ are thus made homogeneous.

The centre of gravity of the disc, axle, and the ring which carries the pivots of the axle is fixed, and the whole is moveable about that centre in any manner, subject to the condition that the line of the pivots of the ring is always horizontal, unless when detached from the stand. Let this straight line of the pivots be denoted by AB, the common centre of the disc and ring by O, the extremities of the axle by N and S; and ON=a.

Let M denote the place of a particle of the disc, its position being determined by the angle AOM (0), and let M' be another point in

the disc indefinitely near to M, but more remote from A, the direction in which the disc will presently be supposed to revolve being AMM'B.

A given force F is applied at N perpendicular to the plane ANBS, so that the disc may describe an angle round AB in the time t; whereby the points M and M' describe the two arcs MP=y and M'P'y' simultaneously. Suppose the circumference of the circle AMB to be divided into four quadrants, commencing at A, where y=0, and corresponding with the four ranges of value of 0 through each of four right angles; suppose M and M' to be in the first quadrant, so that y' is greater than y; then if the disc be supposed to revolve, a particle at M is carried from the line MP to the line M'P', so as to acquire an increase of velocity from the plane AMM' independently of the force F, and consequently (by the first of the two verbal formula) all the momentum so acquired by the particle is lost to the disc, ring, &c., which are thus impelled as by a force in the direction PM or P'M', so as to oppose the rotation imparted by F, but to impart another round O in the direction ANB in the plane of the ring; i. e. in a plane perpendicular to that in which F acts. A force having the same tendency is found, by means of one or the other of the two verbal formulæ, in the other three quadrants, and thus every particle (dm) of the disc contributes to the same effect. This effect is due to the difference of the velocities dy dy' _dy dm lost or and dt

dt

'dy' at P and P', or to the momentum

gained by the particle dm in the time dt.

dy dt

dt dt

is obtained from the equation y=ro sin 0, making
dy'
dt

$ and 0 to vary; but the value of is obtained from that of

The value of

both

dy

dt

by making 0 only to vary. It is thus shown that

It is thence shown, by taking the moments about AB, and applying D'Alembert's principle, that

do fsin 0

Fag

10. cos ode=

dt,

crs

the integrals applying to 0 only, and between the limits O and 2; i. e. to all the particles of the disc simultaneously and independently of p or t. From this is obtained the result

This value being periodical, and ranging between the limits O and shows that the disc makes oscillations which are

the maximum

4Fag Wr2w2

of less extent and duration, as the spinning of the disc is more rapid; i. e. as w is made greater compared with ; and thus if F

F

W

denotes a small weight (such as is usually supplied with the apparatus by the makers), the extent of the oscillation becomes insensible. This formula, applied to the apparatus with which the experiments were made, gives the theoretical maximum of about 18 minutes of a degree. It is evident that when F represents a weight, it should be replaced in the differential equation by F cos 4, but the result practically coincides with that actually obtained when F is not excessive.

That these oscillations must exist will be evident, when it is considered that the gyroscope, with the weight attached and the disc not spinning, becomes an ordinary pendulum: the effect of the spinning being to disturb its oscillations, and to lessen their extent to an unlimited amount, whenever the spinning of the disc is sufficiently rapid.

The preceding investigations, as well as the experiments, show that whenever a force is applied to the axis of a revolving disc, more or less of the momentum due to this force is converted into a momentum of rotation parallel to a plane which is perpendicular to that in which the force acts.

XLVI. Intelligence and Miscellaneous Articles.

ON DEMIDOVITE, A NEW SPECIES OF MINERAL FROM NIJNE
BY N. NORDENSKIOLD.

TAGUIL IN THE URAL.

For some years there has been brought from Taguil* a mineral which occurs but rarely, and to which the name of blue malachite has been given on the spot. The name cannot by any means be applied to the mineral in question, as it contains no traces of carbonate of copper, although it is found in thin layers with malachite. This mineral possessing, as will be seen hereafter, a new and very interesting composition, I venture to propose giving it the name of Demidovite, as a permanent homage to His Excellency M. Anatole de Demidov, who has so powerfully contributed to the advancement of mineralogy and geology by his munificence and his scientific travels.

* These works belong to Prince Anatole Demidoff, and to his nephew Paul Demidoff. They lie 150 wersts (about 100 English miles) to the north of Ekaterinenburg, on the east side of the frontier between Ásia and Europe, but one part of the property is also on the European side.

The yearly produce amounts to more than 1,000,000 pud† of iron (the bar iron from there is known in this country under the name of old Sobel iron); nearly 240,000 pud of copper; and about 50 pud of native platina and 25 pud of gold is yearly washed from the sand. The platinum is only found on the European side of the Ural Mountains, but gold on both sides, although for the most part on the east, or Asiatic side. A great deal of the iron and copper goes directly to Asia.

The forests whence the charcoal is derived are very extensive, and give, when sixty years are allowed for their renovation, 275,000 korob of charcoal yearly, every korob containing 70 cubic English feet.

The proprietors avail themselves of every new discovery in mining and metallurgy, and have in their employment natives of nearly every country of Europe, England, France, Germany, Sweden and Finland.

†The pud is somewhat less than 40 lbs.