Royal Society. 485 



rest is accumulated or treasured up in the moving parts of the ma- 

 chine, and is reproducible whenever 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 the 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 corresponding 

 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 V 2 

 2 u) \-, 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 X 2 is 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 

 coefficient of equable motion. 



He then proceeds to investigate general methods for the determina- 

 tion 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 formulge which results 

 from the introduction of friction, by the ordinary methods, into the 

 consideration of questions of equilibrium, the author calls to bis aid 



