Page:Proceedings of the Royal Society of London Vol 4.djvu/345

Rh its transmission from the moving to the working points ; and all the rest is accumulated or treasured up in the moving parts of the ma- chine, and is reproducible M^henever the work of the moving power from exceeding shall fall short of that which must be expended upon the useful and the prejudicial resistances to carry on the machine.

He then proceeds to observe, that in every machine there thus exists a direct relation between these four elements, — the work done upon the moving points, that expended at the working points, that expended on the prejudicial resistances, and that accumulated in the moving elements. This relation, which is always the same for the same machine, and different for different machines, he proposes to call, in respect to each particular machine, its modulus ; and he states the principal object of this paper (and of another which he proposes subsequently to submit to the Society) to be, first, the general de- termination of the modulus of a simple machine ; secondly, that of a compound machine, from a knowledge of the moduli of its compo- nent elements ; and, thirdly, the application of these general methods of determination to some of the principal elements of machinery, and to tlie machines which are in common use.

The author then states, that the velocities of the different parts, or elements of every machine are connected with one another by certain invariable relations, capable of being expressed by mathematical for- mulae ; so that, though these relations are different for different ma- chines, they are the same for the same machine. Thus it becomes possible to express the velocity of any element of a machine, at any period of its motion, in terms of the corresponding velocity of any other element. Whence it results that the whole vis viva of the machine may at any time be expressed in terms of the corresj)onding velocity of its moving point (that is, the point where the moving power is applied to it), and made to present itself under the form 2 w X^, where V represents the velocity of the moving point of the machine, w the weight of any element, and X a factor determining the velocity of that element in terms of the velocity V of the moving point. Substituting this expression for the vis viva or accumulated work in the modulus and solving in respect to V, an expression is obtained, whence it becomes apparent that the variation of the velo- city V of the moving point, produced by any given irregularity in the work done upon the moving or working points, will be less, as the factor 2 w ig greater. This factor, determinable in every ma- chine, and upon which the uniformity of its action under given variations of the power which impels it depends, he proposes to in- troduce into the general discussion of the theory of machines as the eoefficient of equable motion.

He then proceeds to investigate general methods for the determination of the modulus of a machine, deducing them from those general relations which are established by the principles of statics, between the pressures applied to the machine, in its state bordering upon motion.

That he may escape that complication of formulae which results from the introduction of friction, by the ordinary methods, into the