828 Transactions of the American Institute. 



the crank be taken as radius, the cosines will always be equal to the 

 distance of the reciprocating body from the middle point of its course, 

 it follows that if, upon a line, as OBG-, representing this course, in 



which B is the middle point, 

 rectangular ordinates be drawn, 

 each equal to its distance from B, 

 these ordinates will be propor- 

 tional to the accelerating forces 

 at the points corresponding to 

 them ; and the line drawn 

 through their extremities will 

 be a straight line, showing that 

 the force is uniformly dimin- 

 ishing. 



2. In order to find the law ac- 

 cording to which the force stored 

 up in the piston at mid-stroke is subsequently imparted to the crank, we 

 must first find the expression for this living force and the work it 

 represents, and then take the differential of this stored-up work, which 

 represents its subsequent progressive diminution, on the condition 

 that the crank motion is uniform. 



The work dune in generating the velocity -™— during the first half 



stroke, is found by integrating the expression, 



(2-) 2 

 dW = Fds = -W- ; rcos.^mn.^d^, 



which gives W = — &W ?' 2 cos.V -f C, the mass acted on being 



regarded as unity : where "W represents work in the piston. 

 Now when ?=0°, "W=0, and cos.?' =1. 



"Whence W = -jrM r^l — cos. 2 ?) = ^w 7 ^ sin,V 



When <p = 90°, sin. ? =l and W=4S$V*. 



In the second quadrant this operation is simply reversed, thus: 



dW =:~^-r Bin.frcos.fdf = Fds = — Frzm.fd? ; 



and dividing by r sm.fdy. 



-^ rcos.p«= — F, 



the negative sign, showing that the direction of the force is how 

 reversed. This negative character belongs implicitly to the first 



