832 Transactions of the American Institute. 



the same amount of work per stroke with the cut-off k, as is done 

 with full head of steam by the smaller pressure p, by means of the 

 formula : 



2n 



Pr (l+hl^) = 2py 



2r being the length of the stroke, since r = length of crank. 



2pt* 

 Hence P— 



*(i+m») 



If, for simplicity, we put r = 1, we may find the value of P in 

 terms of p for any cut-off, by substituting the proper value for k, 

 remembering that, as r == 1 the whole stroke = 2 ; and that if ~k be 

 taken, as usual, at a fraction of the whole stroke regarded as unity, 

 this fraction must be doubled when introduced into the present for- 

 mula. 



A favorable cut off for the high speed engine would be one-eighth, 

 which would give us 



p = i(i + 2 h.i.8) = jk = 2 - 6p > nearl y- 



In order, therefore, that a cylinder engine, working with a one- 

 eighth cut-off, may do the same amount of work per stroke which the 

 same engine would do working without cut-off under the given pres- 

 sure of steam, p, an initial pressure must be employed two and six- 

 tenths as great as p. 



In an engine in which the reciprocating parts are without inertia, 

 the distribution of work upon the different parts of the stroke with 

 this cut-off is very unequal ; more than three and a half times as 

 much being done in the first quarter revolution as in the second. But, 

 by the use of a heavy piston, this inequality may be reduced to any 

 extent ; and even, as will be obvious from what has gone before, may 

 be entirely reversed. 



In fig. 5, we have seen that the line P' Y may be so drawn as to 

 make the areas P P' B C, C B Y' H' equal to each other. We may 

 easily find the point P' through which it should be drawn, by con- 

 sidering that P' O B = B G Y' ; and that P P' B C = P O B C - 



in which p is a pressure empirically determined, the values of which, for con- 

 densing and for non-condensing engines, are given in his treatise. 



To have employed either of these formulas in the expressions used in this paper^ 

 would only have increased their complexity without materially affecting the conclu- 

 sions arrived at. 



