174 APPENDIX. 



By differentiating this expression of the distance traversed by C, we obtain the velocity of the 

 extremity C, viz. : 



dx odd i p cos. a \ 



-r-. j-T- sin. a I ' .== + 1 ) 5 that is, 



* v v r* p 2 sin. 2 



/ p COS. a v 



v = v sin. a ( . -L i ) (2). 



\ \A^- p 2 sin. 2 a 



But if P denote the moving force acting at C, in the direction C B, P' the effective force of D, 

 and r = n p, it has been shown at page 229 of the present work, by means of the resolution 



of the forces, that P' = P sin. a ( / ' . = +1 ) or, replacing n by -, 



\ V w 2 sin. 2 * J p' 



P' = Psin. a ( /2 PC S ; g . . + 



\ > r * _ 2 sm> 2 a 



p/ 



Equation (2) hence reduces to v v' , and therefore 



that is, the moving power at C is always equal to the effective power at D. 



The moving force P being nearly uniform, the force P* expressed by the equation (3) will 

 be materially different at different positions of the crank, and the velocity of the engine will, 

 in consequence, be subject to small fluctuations in the course of each revolution. To 

 ascertain the law of these variations, it will be necessary to have recourse to the usual equation 

 of rotatory motion. Let R denote the real moment of the resistance, including friction, 

 reduced to the point D, or the force which uniformly applied along the circumference E D 

 would just suffice to preserve the mean velocity of the engine without any variation. Then 

 being the force actually applied, the effective accelerating force, or the part tending to pro- 



d a. 

 duce acceleration, will be P 7 R ; and where this is negative, the velocity - must be re- 



'' / 



tarded instead of accelerated. If we now multiply the force P' R by p, the leverage at 

 which it acts, the product p (F R) is its moment or tendency to generate angular motion. 

 Hence, if M designate the moment of the inertia to be overcome, and if we neglect, as com- 

 paratively insignificant, the slight variations of the resistance R due to the small changes of 

 velocity, and suppose it to continue uniform, we shall have, 



M ? = P ( P/ ~ R )' 



Multiply by -1, and 



M " = 2 P' p - 2 R p 



dt 3 v dt v dt 



2 PV - 2 R p 



f dt 



2Pv - 2Rp p. 



(' > 



n -T. dx n r, da. 



