tested them by other known experiments, and especially refers to those by Major Wade on the transverse strength of square and round bars of cast iron of different qualities, related in the "Reports on the Strength and other Properties of Metals for Cannon," presented to the United States Government by the Officers of the Ordnance Department. The unit of strength, as computed by Major Wade from these experiments, came out uniformly much higher in the round than in the square bars of the same kind of iron, whence he was led to doubt the correctness of the formula employed; but the author shows that when his formula is used, which includes the resistance of flexure, the discrepancy referred to disappears, and the tensile resistance, whether obtained for the round or the square bars, agrees very nearly with that derived from the experiments on direct tension under like circumstances.

As to the ratio between the resistance of flexure and the tensile resistance, it is remarked that, were the metal homogeneous, the former resistance would probably be precisely equal to the latter, instead of bearing the ratio of nine-tenths, as found by experiment; but the ratio evidently varies in different qualities of metal; and accordingly from Major Wade's experiments, it appears that with the same metal subjected to different modes of casting, an increase of transverse strength may accompany a decrease in the tensile resistance.

Respecting the limit of action of the resistance of flexure, the author observes, that in all the simple solid sections, the points of action are evidently the centres of gravity of the half-section; while in the compound sections it is necessary to compute the centre rib and flanges as for two separate beams in which the resistance of flexure is different, and has its point of action at the centre of gravity of the separate portions. It would appear that the elastic reaction developes this resistance to the full extent when the section is such that a straight line may be drawn from every point at the outer portion to every point at the neutral axis within the section; but that if the form of section is such that straight lines drawn from the outer fibres or particles to the neutral axis fall without the section, then it must be treated as two separate beams, each having that amount of resistance of flexure due to the depth of the metal contained in it.

The last section of the paper is devoted to the consideration of the resistance of flexure in wrought iron; and experiments are first

given to determine the position of the neutral axis, from which it is found to be at the centre of gravity of the section, as in cast iron; so that the action is the same in both materials, except as to the amount of the extensions and compressions with a given strain; and the formulæ given for cast iron will also apply to wrought iron. As wrought iron yields by bending and not by fracture, the relative value off and are not so easily ascertained; moreover the ultimate compressive strain which wrought iron can sustain is little more than half its ultimate tensile strength; nevertheless the force required to overcome the elasticity of the material is nearly the same, whether applied as a compressive or tensile strain; the difference being, that the force which overcomes elasticity when applied as a compressive strain leads to the destruction or distortion of the material, while, in the case of the tensile strain, the elasticity may be overcome long before the material yields by absolute rupture.

A statement is given of the results of experiments made by Professor Barlow, in 1837, to show the weights which overcome the elasticity of the metal when applied transversely as compared with the weight necessary to produce the same result when applied by direct tension, and from these it is concluded that the resistance of flexure in wrought iron, considered as a force acting evenly over the surface, is nearly equal to one-half of the tensile resistance.

In an Appendix to this paper, by Professor Barlow (read at the following meeting), the preceding principles are applied to beams and rafters of non-symmetrical section.

With this view, the case of the double-flanged girder with unequal flanges is selected and discussed, and formulæ deduced, which are then tested by comparison with the results of experiments by Prof. Hodgkinson, published in the Manchester Memoirs ;' a selection being made of those in which the girders differed most from each other in section, dimensions, and bearing-distance. The chief particulars of these experiments are given, with diagrams showing the forms of sections, and the values as obtained from the formulæ are stated. The value of the direct tensile strength of cast iron thus derived, falls between the limits of 1400 and 1700.

In the Reports of the Commissioners of Inquiry into the "Application of Iron to Railway Structures," are given the results of about fifty experiments on the direct tensile resistance of one-inch square

cast-iron bars, under the direction of Professor Hodgkinson. The bars consisted of seventeen different kinds of iron, each set of bars being of the like quality and manufacture; and in several of these sets, which might have been expected to yield the same results, the difference is fully as great as in the cases here exhibited. From this fact an inference may be drawn in favour of the general applicability of the principles developed in the foregoing pages to cast-iron beams and girders of every variety of section.

II. "On the Theory of the Gyroscope." By the Rev. WILLIAM Cook, M.A. Communicated by Professor A. WILLIAMSON, P.R.S. Received February 13, 1857.

(Abstract.)

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.

I. 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, 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

}

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

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 for

mulæ, masses are represented by weights; so that any effective accelerating force ƒ 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 Τ ; 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 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 formulæ) 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

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

dy'

dt

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

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

dt

аф (dy - dy) dm=(c

dt dt

cos 0. -w. sin
dt

0) rw

rwdtdm.

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

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

This value being periodical, and ranging between the limits O and the maximum

4Fag Wr2w2

shows that the disc makes oscillations which are

of less extent and duration, as the spinning of the disc is more F W

rapid; i. e. as w2 is made greater compared with ; and thus if F

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.