P OLYTECHNIC ASSOCIATION: 831 



Suppose, for instance, the steam is cut off at mid stroke, the diagram 

 of work will be such as is here shown : 



The rectangle cor- 

 responding to the 

 work of the second 

 half stroke is reduced 

 by the area C H H', 

 but the remaining 

 area, CBY IP, still 

 exceeds the triangle 

 P B C, which repre- 

 sents the work of the 

 first half stroke. Let, 

 now, through the 

 point, B; another line 

 be drawn, as P' B Y', and suppose such a weight given the reciprocating 

 parts of the engine that P' O may suffice to impart to them the neces- 

 sary initial acceleration, then PP'BC will be the diagram of work 

 for the first quadrant, and CBY' II' the corresponding diagram for 

 the second quadrant. It is obvious that such a position may be 

 found for P' Y' as to make these two diagrams equal. 



It is to be noticed, however, that if we wish the engine to perform, 

 when worked with cut-off, the same amount of work which we have 

 supposed it to perform when working with full head of steam (viz. : 

 that represented by the rectangle P O G II), the initial pressure 

 will have to be increased to a degree which is a function of the part 

 of stroke performed before cut-off. And as it is usual to compute 

 roughly the work of expansively working engines by means of 

 hyperbolic logarithms,* we may find the pressure P, which would do 



* If it were important to the object of this inquiry that the elastic force of expand- 

 ing' steam, and the work done during exhaustion, should be very exactly stated, it 

 would hardly be proper to depend, in this part of the investigation, upon the hyper- 

 bolic theory, or the law of Marriotte. The formula of Poisson for expanding 

 gases, viz. : 



though not strictly true of vapors which, like steam, partially condense during 



exhaustion, would give results nearer the truth. Pambour's formula, now generally 



accepted, is the formula of Marriotte, with a negative term annexed, increasing with 



the exhaustion ; and may be thrown under this form : 



k x — k; 

 P — P — ■ — p 



x * x 



