laws of friction. In making this deduction I have, in every case, availed myself of the following principle, first published in my paper on the theory of the arch, read before the Cambridge Philosophical Society in Dec. 1833, and printed in their Transactions of the following year:-"In the state bordering upon motion of one body upon the surface of another, the resultant pressure upon their common surface of contact is inclined to the normal, at an angle whose tangent is equal to the coefficient of friction."

This angle I have called the limiting angle of resistance. Its values calculated, in respect to a great variety of surfaces of contact, are given in a table at the conclusion of the second part, from the admirable experiments of M. Morin *, into the mechanical details of which precautions have been introduced hitherto unknown to experiments of this class, 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

* Nouvelles Expériences sur le Frottement, Paris, 1833. † In my memoir on the " Theory of Machines" (Phil. Trans. 1841), I have extended this relation to the case in which the num

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

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

to wheels having epicycloidal 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 have called it the modulus of stability *; conceiving this 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,

*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 is ths the distance between the extrados at the base and the vertical through the centre of gravity.

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 form.

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 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.

* 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.