PROFESSOR MOSELEY ON THE THEORY OF MACHINES. 
291 
space Sj traversed by the point of application of the moving power ; so that, repre- 
Vj 
senting Pj by its value - 1 , we have by equation 3, 
*1 
F (1*25 & 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 ^ repre- 
sents in this case the projection of the increment A Si of the space described by the 
point of application of Pi on the direction of that force, we have s Y = A Sj cos 0 ; 
observing, therefore, that u x is the increment of U l5 and representing it by A U l5 we 
have 
u x AUj 
~ 
1 
cos 6 
= F ( p 2 > P 3 • &C 0 > 
and passing to the limit 
d Uj 
dS x 
= cos 0 . F (P 2 , P 3 , &c.). 
. .U l = /cos4.F(P 2 ,P 3 , &c.) dS, (5.) 
where 0 and F (P 2 , P 3 , &c.) are functions of S L . 
The work \J l done through a given space S : at the driving point under the pres- 
sures P 2 , p 3> &c., at the working points of the machine, is determined by this equa- 
tion in terms of S P Now the pressure P 2 is given in terms of the work U 2 done by 
it, and the distance S 2 through which it is done ; and S 2 is given in terms of Sj ; so 
that P 2 is given in terms of U 2 and S x . In like manner P 3 is given in terms of U 3 
and Sj ; and so of the rest. If, therefore, we substitute for P 2? P 35 &c. in the above 
equation their values thus determined, we shall obtain a relation between U l5 U 2 , U 3 , 
&c. and S l5 which is the modulus required. 
6. There exists in every case a relation between the quantities p 2 , p 3 , &c., which 
will be found useful in determining the moduli of a large class of machines. Let 
P^ 05 be taken to represent that value of Pj 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 (0) (P 2 , P 3 , &c.) represent the corresponding value of F (P l5 P 2 , &c.), 
Pi® = F™ (P» P 3 , &c.). 
Also by the principle of virtual velocities, since P/ 0) , P 2 , P 3 , &c. are pressures in equi- 
librium, we have 
Pi (0) • s i — P 2 • ^2 + P3 * S 3 + — ; 
substituting for s 2 , s 3 , &c., their values — > — , &c., and dividing by % 
f*2 P'3 
MDCCCXLI. 
f, + S + &c ' = F ' 0> < p » p * &c -> 
2 Q 
( 6 .) 
