290 PROFESSOR MOSELEY ON THE THEORY OF MACHINES. 



the pressures upon its different working points, and let that relation which obtains at 

 any period of the motion between the moving pressure P^ and the working pressures 

 Pg, P3, &c., when in the state bordering upon motion, and subject to the various 

 prejudicial resistances under which the machine works, be represented by 



Pi = F(P2,P3,&c.) (3.) 



Let *i, ^2J ^3i &c. represent the spaces described in the same exceedingly small time 

 by the points of application of P^, Pg, &c., if these points move in the directions in 

 which those pressures severally act, and if not let them represent the projections of 

 these spaces on the directions of the pressures. Then are these spaces, S2, s^, &c., evi- 

 dently related to the space Aj by equations of the form 



^2 "^2 = hi f*3 *3 = *1> ^4 *4 = -^IJ &C. &C., 



where jU/gj i^3> (^a ^^' ^^^ certain constant quantities determined by the forms and di- 

 mensions of the moving elements of the machine and their combination, or certain 

 functions of these and of the space s^ which the moving point has described from 

 the commencement of any given period of its motion. Let now u^ represent the 

 work of the pressure Pj through the space Si, u^ that of Pg through $2, &c. 



.'. U^ = "1 -^1, 2/'2 ^^ "2 '^2' % ^^ "3 *35 C^G'. 

 .-. Pl = ^ P2=:^^^, P3 = ^&C. 



••■:i = F(«^'^,&e.) (4.) 



Which equation, — expressing a relation between the work u^ at the driving point, 

 through a small increment s^ of the space Sj described by that point, and the work 

 Wg, W3, &c. yielded during the same period at the several working points — is the 

 modulus of the machine in respect to an exceeding small motion of its elements. 



If the pressures Pj, Pg, &c. remain constant during any given period of the opera- 

 tion of the machine, and act continually in the same directions, it is evident that the 

 above reasoning obtains whatever may be space s^ through which the work u^ is done ; 

 so that the exceeding small quantities u^, u^^ &c. s-^ may in this case be replaced by the 

 finite quantities Uj, Ug, &c. Sj*; S^ representing any finite space through which the 

 work Uj is done at the driving point, whilst the work Ug, U3, &c. is yielded at the 

 working points of the machine. 



If the pressures Pj, P2, P3, &c. be variable during any given period of the continuous 

 operation of the machine, as it respects their several amounts, or their directions, or 

 as to both these elements, then are they (in every case presented in the operation of 

 machinery, simply and without the interposition of any voluntary agent) functions of 

 the spaces S^, Sg, S3, &c. traversed by their points of application, and therefore of the 



* If the direction of the pressure P, be other than that in which its point of application is made to move, S, 

 must be taken to represent the projection of the space described by that point on the direction of the force. 



