PROFESSOR MOSELEY ON THE THEORY OF MACHINES. 291 



space Sj traversed by the point of application of the moving power ; so that, repre- 

 senting Pi by its value j, we have by equation 3, 



^ = F (Pj, P3, &c.), 



where the second member is a function of Sj. Now if the direction in which the 

 point of application of Pj is made to move do not coincide with the direction in which 

 that force acts, being inclined to it in any position at an angle 6, then, since s^ repre- 

 sents in this case the projection of the increment A Sj of the space described by the 

 point of application of Pj on the direction of that force, we have *i = A Sj cos 6 ; 

 observing, therefore, that u^ is the increment of Uj, and representing it by A Uj, we 

 have 



and passing to the limit 



^=C0S^.F(P2,P3,&C.). 



.-. Ui=y*cos^.F(P2,P3, &c.) rfSi ....... (5.) 



where 6 and F (Pg, P3, &c.) are functions of Sj. 



The work Uj done through a given space Si at the driving point under the pres- 

 sures P2, P3, &c., at the working points of the machine, is determined by this equa- 

 tion in terms of Si- Now the pressure Pg is given in terms of the work U2 done by 

 it, and the distance Sg through which it is done ; and Sg is given in terms of Si ; so 

 that P2 is given in terms of Ug and Si- In like manner P3 is given in terms of U3 

 and Si ; and so of the rest. If, therefore, we substitute for Pg, P3, &c. in the above 

 equation their values thus determined, we shall obtain a relation between Ui, Ug, U3, 

 &c. and Si, which is the modulus required. 



6. There exists in every case a relation between the quantities jo-g, /"'s, &c., which 

 will be found useful in determining the moduli of a large class of machines. Let 

 V^^^ be taken to represent that value of Pi which would be necessary to give motion 

 to the machine if there were no prejudicial resistances opposed to the motion of its 

 parts ; and let F^<^^ (Pg, P3, &c.) represent the corresponding value of F (Pi, Pg, &c.), 



.-. P/o) = FO) (P2, P3, &c.). 



Also by the principle of virtual velocities, since Pi ^^^, Pg. P3, &c. are pressures in equi- 

 librium, we have 



"1 • *1 = "2 • *2 "T* "3 • *3 "T • • • • » 

 S 8 



substituting for s^, *3, &c., their values — » — , &c., and dividing by a'i, 



J + & + &c. = F'O' (P^, P3, &c.) (6.) 



MDCCCXLI. 2 Q 



