Polytechnic Association. 829 



member of the equation, also, since cos. <p is negative in the second 

 qnadrant. 



The law of retardation in the second half stroke is, therefore, the 

 same as the law of acceleration in the first. 



The expression for work found above, viz. : 



-^jgV sin.V 5 or, generally, -^V sm.V - 



(in which w is the weight of the reciprocating body, and g, the force 

 of gravity), represents the amount of work which continues to b& 

 stored up at the successive points of the second half stroke at which 

 s — v. s. 9; and, therefore, the amount which will have been, at 

 these successive points, transferred to the crank, will be expressed by 



TVT, (2 7r) 2 „ W M • 2 \ ( 27r ) 2 2 W 2 



W = ^ 2 r 2 - (1 -sin.V) = -^ r 2 - cos. 3 <p. 



At the end of the stroke, 9 '— 180°, sin. 9 = 0, and cos. 9 = — 1. 



Here W = and W = V^ ^ — ; or, the living force of the 



2r 9 



moving mass has been entirely exhausted, and its equivalent of work 



transferred to the crank. 



This case is illustrated in figure three. Producing the line P B to 

 meet G Y, drawn perpendicularly to BG downward, the ordinates 

 TJ Y, "W X, parallel to Gr Y, are proportional to the forces exerted on 

 the crank in a direction parallel to O G at the points U V of the 

 path of the piston ; and the triangle JB G Y represents the work done 

 by these forces during the last half stroke. 



3. The third of the questions above proposed relates to the law 

 according to which the force of the steam reaches the crank during 

 the first half stroke in presence of the heavy piston, on supposition 

 that the pressure in the cylinder is constant. This is substantially 

 answered in the solution already given to the first question. If the 

 reciprocating parts of the engine were wholly without inertia, the 

 diagram of work done would be a rectangle, the ordinates represent- 

 ing the force acting on the crank in a direction parallel to the line of 

 the centers being all equal. 



