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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

ог

(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. J

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+F' ̄ ̄E'm+F'n+C'

am tôn ta=ôm+n+p=em+on+y=0.

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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+By2+Cz2+G=0. Moreover, the system of axes is now rectangular: hence the axis of y, and that of z, equally with that of x, 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, 62, 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+A,
Q=Dx+By+F+B2

R=Ex+Fy+Cz + C2
S=A+By+Cq≈ + G

Then Pr+Qy+R+S=0 is the eq. to the surface, and Pa'+Qy' +R+S=0 is the eq. to the tangent plane at (xyz). Hence if

η

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, Re'+S=0. Let Enz be the reciprocals of x'y' z'. Then P+S=0, Q+nS=0, R+8=0; and the eq. to the surface becomes Ex+ny +-1=0. Restore for PQR their equivalents; then eliminating xyzS you get

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

a2+bn2+ch2+2a‚§+2b2n+2c25+2d&n + 2e§5+2ƒn$+g=0,

it is not difficult to obtain a system of eqq. in which abc...En play the same part, as just before did ABC...xyz. Whence again we have

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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 tangential 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, 1857.

Major-General SABINE, R.A., Treas. and V.P., in the Chair.

The following communications were read :—

I. "On an Element of Strength in Beams subjected to Transverse Strain, named by the author The Resistance of Flexure."" (Second Communication.) By WILLIAM HENRY BARLOW, Esq., F.R.S. Received March 12, 1857.

(Abstract.)

In his former paper on this subject the author pointed out the existence of an element of strength in beams when subjected to transverse strain, the resistance of flexure-which had been omitted in the generally received theory; and the object of the present experimental inquiry is to elucidate more clearly the general bearing of the subject, and determine more precisely the laws which govern this resistance.

The forms of beam employed in the experiments formerly described were only of two kinds-solid rectangular bars and open girders; in the present experiments other forms have been used, namely, square bars broken on their sides, square bars broken on their angles, round bars, beams of the I section broken with the flanges horizontal, and similar beams broken with the flanges vertical H.

The results of these experiments are exhibited in Tables, together with those of the former series; and the author employs them, in the first place, to test the accuracy of the existing theory, by comparing the resistance of the outer fibres or particles of each of the forms of beam, calculated on that theory, with the actual tensile strength of the metal as obtained by direct experiment. From this comparison applied to the different forms of beam, it would follow that the resistance at the outer fibre varies from 25,271 lbs. to 53,966 lbs., while the tensile strength of the metal, obtained by experiments on direct tension, averages only 18,750 lbs. ; and the dis

crepancy and variation will be found to arise from the received theory not taking into account the resistance consequent on the molecular disturbance accompanying curvature.

In his former paper the author gave a formula by which the difference between the tensile strength and the apparent resistance at the outer fibre could be computed, approximatively, in solid rectangular beams and open girders; and he now proposes to trace the operation of the resistance of flexure, considered as a separate element of strength, and to show its effect, in each of the forms of section above indicated. Observing that the usual supposition of only two resistances in a beam, tension and compression, fails to account either for the strength, or for the visible changes of figure which take place under transverse strain, he proceeds to discuss the effects involved in such change of figure, and thence arrives at the following conclusions applicable to the resistance of flexure :—

1. That it is a resistance acting in addition to the direct extension and to compression.

2. That it is evenly distributed over the surface, and consequently (within the limits of its operation) its points of action will be at the centres of gravity of the half-section.

3. That this uniform resistance is due to the lateral cohesion of the adjacent surfaces of the fibres or particles, and to the elastic reaction which thus ensues between the portions of a beam unequally strained.

4. That it is proportional to and varies with the inequality of strain, as between the fibres or particles nearest the neutral axis and those most remote.

Formulæ are then given, according to these principles, exhibiting the relation between the straining and resisting forces in the several forms of section experimented on, as resulting from the joint effect of the resistances of tension, compression and flexure. The application of these formula to the actual experiments yields a series of equations with numerical coefficients, in which, were the metal of uniform strength, the tensile strength f, and the resistance of flexure , would be constant quantities, and their value might be obtained from any two of the equations; but as the strength varies even in castings of the same dimensions, and as a reduction of strength per unit of section takes place when the thickness is increased, the values

of ƒ and will necessarily vary, and can only be ascertained in each experiment by first establishing the ratio they bear to each other. For this purpose the first ten experiments are used, in all of which the metal was from to 1 inch in thickness, and its mean tensile strength ascertained by direct experiment to be 18,750 lbs. per inch. The resulting mean value of $ is=16,573 lbs., and the ratio of ƒ to as 1 to 847.

By using results obtained by Prof. Hodgkinson on the breaking weight of inch bars of ten different descriptions of iron, where the tensile strength was ascertained by direct experiment, it would appear that the ratio between the resistance of tension and the resistance of flexure varies in different qualities of metal, an inference which seems to be confirmed by other experiments on rectangular bars given in the Report of the Commissioners on the application of iron to railway structures. The mean result, however, accords nearly with that of the author's experiments, and gives the ratio of f to as 1 to 853. Hence, according to these data, the resistance to flexure, computed as a force evenly distributed over the section, is almost nine-tenths of the tensile resistance.

This ratio of the values of ƒ and 4 being applied to the equations resulting from the several experiments, gives the tensile strength of the metal as derived from each form of section, and the results, though not perfectly regular, are found to be within the limits of the variation exhibited by the metal as shown by the experiments on direct tension in the former paper. Classified and condensed, these results are as follows:

The mean tensile strength as obtained from

The open girders, is....

18,282

The solid rectangular bar of 2 inches sectional area 17,971
The inch bars-square and round, and square broken.

diagonally..

19,616

The bars of 4 inches sectional area, square and
round, and square broken diagonally ..... 16,800
The compound sections in which the metal was

inch thick

19,701

Having thus found that his formulæ, when applied to his own experiments, gave consistent and satisfactory results, the author next

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