and which have given to our knowledge of the laws of friction a precision and a certainty hitherto unhoped for.

Of the various elements of machinery those which rotate about cylindrical axes are of the most frequent occurrence and the most useful application; I have, therefore, in the first place sought to establish the general relation of the state bordering upon motion between the driving and the working pressures upon such a machine, reference being had to the weight of the machine.* This relation points out the existence of a particular direction in which the driving pressure should be applied to any such machine, that the amount of work expended upon the friction of the axis may be the least possible. This direction of the driving pressure always presents itself on the same side of the axis with that of the working pressure, and when the latter is vertical it becomes parallel to it; a principle of the economy of power in machinery which has received its application in the parallel motion of the marine engines known as the Gorgon Engines.

I have devoted a considerable space in this portion of my work to the determination of the modulus of a system of toothed wheels; this determination I have, moreover, extended to bevil wheels, and have included in it, with the influence of the friction of the teeth of the wheels, that of their axes and their weights. An approximate form of this modulus applies to any shape of the teeth under which they may be made to work correctly; and when in this approximate form of the modulus the terms which represent the influence of the friction of the axis and the weight of the wheel are neglected, it resolves itself into a well known theorem of M. Poncelet, reproduced by M. Navier and the Rev. Dr. Whewell. In respect to wheels having epicy

In my memoir on the "Theory of Machines" (Phil. Trans. 1841), I have extended this relation to the case in which the number of the pressures and their directions are any whatever. The theorem which expresses it is given in the Appendix of this work.

In the discussion of the friction of the teeth of wheels, the direction of the mutual pressures of the teeth is determined by a method first applied by me to

cloidal and involute teeth, the modulus assumes a character of mathematical exactitude and precision, and at once. establishes the conclusion (so often disputed) that the loss of power is greater before the teeth pass the line of centres than at corresponding points afterwards; that the contact. should, nevertheless, in all cases take place partly before and partly after the line of centres has been passed. In the case of involute teeth, the proportion in which the arc of contact should thus be divided by the line of centres is determined by a simple formula; as also are the best dimensions of the base of the involute, with a view to the most perfect economy of power in the working of the wheels.

The greater portion of the discussions in the third part of my work I believe to be new to science. In the fourth part I have treated of "the theory of the stability of structures," referring its conditions, so far as they are dependent upon the rotation of the parts of a structure upon one another, to the properties of a certain line which may be conceived to traverse every structure, passing through those points in it where its surfaces of contact are intersected by the resultant pressures upon them. To this line, whose properties I first discussed in a memoir upon "the Stability of a System of Bodies in Contact," printed in the sixth volume of the Camb. Phil. Trans., I have given the name of the line of resistance; it differs essentially in its properties from a line referred to by preceding writers under the name of the curve of equilibrium or the line of pressure.

The distance of the line of resistance from the extrados of a structure, at the point where it most nearly approaches it, I have taken as a measure of the stability of a structure,* and

that purpose in a popular treatise, entitled Mechanics applied to the Arts, published in 1834.

*This idea was suggested to me by a rule for the stability of revêtement walls attributed to Vauban, to the effect, that the resultant pressure should intersect the base of such a wall at a point whose distance from its extrados ie ths the distance between the extrados at the base and the vertical through the centre of gravity.

have called it the modulus of stability; conceiving thie measure of the stability to be of more obvious and easier application than the coefficient of stability used by the French writers.

That structure in respect to every independent element of which the modulus of stability is the same, is evidently the structure of the greatest stability having a given quantity of material employed in its construction; or of the greatest economy of material having a given stability.

The application of these principles of construction to the theory of piers, walls supported by counterforts and shores, buttresses, walls supporting the thrust of roofs, and the weights of the floors of dwellings, and Gothic structures, has suggested to me a class of problems never, I believe, before treated mathematically.

I have applied the well known principle of Coulomb to the determination of the pressure of earth upon revêtement walls, and a modification of that principle, suggested by M. Poncelet, to the determination of the resistance opposed to the overthrow of a wall backed by earth. This determination has an obvious application to the theory of foundations.

In the application of the principle of Coulomb I have availed myself, with great advantage, of the properties of the limiting angle of resistance. All my results have thus received a new and a simplified forın.

The theory of the arch I have discussed upon principles. first laid down in my memoir on "the Theory of the Stability of a System of Bodies in Contact," before referred to, and subsequently in a memoir printed in the "Treatise on Bridges" by Professor Hosking and Mr. Hann.* They differ essentially from those on which the theory of Coulomb is founded; when, nevertheless, applied to the case treated

I have made extensive use of the memoir above referred to in the following work, by the obliging permission of the publisher, Mr. Weale.

The theory of Coulomb was unknown to me at the time of the publication of my memoirs printed in the Camb. Phil. Trans. For a comparison of the two methods see Mr. Hann's treatise.

by the French mathematicians, they lead to identical results. I have inserted at the conclusion of my work the tables of the thrust of circular arches, calculated by M. Garidel from formulæ founded on the theory of Coulomb.

The fifth part of the work treats of the "strength of materials," and applies a new method to the determination of the deflexion of a beam under given pressures.

In the case of a beam loaded uniformly over its whole length, and supported at four different points, I have determined the several pressures upon the points of support by a method applied by M. Navier to a similar determination in respect to a beam loaded at given points.*

In treating of rupture by elongation I have been led to a discussion of the theory of the suspension bridge. This question, so complicated when reference is had to the weight of the roadway and the weights of the suspending rods, and when the suspending chains are assumed to be of uniform thickness, becomes comparatively easy when the section of the chain is assumed so to vary its dimensions as to be every where of the same strength. A suspension bridge thus constructed is obviously that which, being of a given strength, can be constructed with the least quantity of materials; or, which is of the greatest strength having a given quantity of materials used in its construction.†

The theory of rupture by transverse strain has suggested a new class of problems, having reference to the forms of girders having wide flanges connected by slender ribs or by open frame work: the consideration of their strongest forms leads to results of practical importance.

In discussing the conditions of the strength of breastsummers, my attention has been directed to the best positions of the columns destined to support them, and to a comparison

As in fig. p. 487. of the following work.

That particular case of this problem, in which the weights of the suspending rods are neglected, has been treated by Mr. Hodgkinson a the fourth vol. of Manchester Transactions, with his usual ability. He has not, however, suc ceeded in effecting its complete solution.

of the strength of a beam carrying a uniform load and sup ported freely at its extremities, with that of a beam similarly loaded but having its extremities firmly imbedded in masonry.

In treating of the strength of columns I have gladly replaced the mathematical speculations upon this subject, which are so obviously founded upon false data, by the invaluable experimental results of Mr. E. Hodgkinson, detailed in his well known paper in the Philosophical Transactions for 1840.

The sixth and last part of my work treats on "impact;" and the Appendix includes, together with tables of the mechanical properties of the materials of construction, the, angles of rupture and the thrusts of arches, and complete elliptic functions, a demonstration of the admirable theorem of M. Poncelet for determining an approximate value of the square root of the sum or difference of two squares.

In respect to the following articles of my work I have to acknowledge my obligations to the work of M. Poncelet, entitled Mécanique Industrielle. The mode of demonstration is in some, perhaps, so far varied as that their origin might with difficulty be traced; the principle, however, of each demonstration-all that constitutes its novelty or its valuebelongs to that distinguished author.

30,* 38, 40, 45, 46, 47, 52, 58, 62, 75, 108,† 123, 202, 267, 268, 269, 270, 349, 354, 365.§

* The enunciation only of this theorem is given in the Méc. Ind., 2me partie, Art. 38.

+ Some important elements of the demonstration of this theorem are taken from the Mec. Ind., Art. 79. 2me partie. The principle of the demonstration is not, however, the same as in that work.

In this and the three following articles I have developed the theory of the fly-wheel, under a different form from that adopted by M. Poncelet (Méc. Ind., Art. 56. Sme partie). The principle of the whole calculation is, however, taken from his work. It probably constitutes one of the most valuable of his contributions to practical science.

§ The idea of determining the work necessary to produce a given deflection of a beam from that expended the compression and the elongation of its component fibres was suggested by an observation in the Méc. Ind., Art. 75. 3me partie.