246 Mr. W. J. M. Rankine on the Mechanical Action of Heat. 



reduced to unity of weight of steam, is 



And the useful effect of uuity of weight of steam being 



A'i(P,z-r.), 



the problem is to determine the ratio of expansion s, so that 



V.(PiZ-F^) 



h+k^ 

 In 



shall be a maximum. 



Dividing the numerator of this fraction by ViPj, and the de- 



kY 

 nominator by -y-', both of which are constants in this problem, 



we find that it will be solved by making the ratio 



Z-f* 



H^ (^^) 



a maximum. 



The algebraical solution would be extremely complicated and 

 tedious. The graphic solution, on the other hand, is very simple 

 and rapid, and sufficiently accurate for all practical purposes, and 

 I have therefore adopted it. 



In the annexed diagram, Plate III. fig. 1, the axis of abscissae, 

 — XO + X, is graduated from towards +X into divisions 

 representing ratios of expansion, or values of s. The divisions 

 of the axis of ordinates, OY, represent values of Z. The curve 

 marked " locus of Z " is laid down from the third column of 

 Table II. of the Appendix to the original paper, being applicable 

 to initial pressures not exceeding four atmospheres. 



Through the origin draw a straight line BOA, at such an 

 inclination to —XO+X that its ordinates are represented by 



F 



:5- s. Then the ordinates measured from this inclined line to 



t^i F 



the locus of Z represent the value of the numerator Z— p-*, of 



the ratio (62), corresponding to the various values of s. 



Take a point at C on the line BOA, whose abscissa, measured 



along — X, represents — y^. Then the ordinates, measured 

 from BOA, of any straight line drawn through C, vary propor- 

 tionally to the denominator rrr- +s of the ratio (62). 



