DAVIS. — A PQ PLANE FOR THERMODYNAMIC CYCLIC ANALYSIS. 045 



boundary of the FT' area as the top of the figure. In general, an effi- 

 ciency curve comes in along the Q axis more horizontally than the V 

 curves, and if followed upward far enough, swings completely back to 

 Q =. CO again, cutting each F curve twice if at all. There will always 

 be one efficiency curve that just touches a given velocity curve. Call 

 the point of tangency S. Its cycle has evidently a higher efficiency than 

 any other cycle giving the same final velocity, for the efficiency de- 

 creases as a point moves away from S in either direction along the given 

 V curve. Therefore, if the point S of the curve V= Flies on the 

 boundary of a given V T^ area, it is the best point of the area for effi- 

 ciency. But if T.\ is so small as to bring the corresponding V T^ corner 

 farther up the F curve than the point S for that curve, then the V T^ 

 corner itself is the best point of the area for efficiency, as it lies nearest 

 ♦S". The locus of the point S for all possible values of F is the line 

 drawn with long dashes in Figure 13. It looks somewhat like one of 

 the T, family, but its equation is quite different.* 



* The analytic work is as follows. The equation of the F curves is 



vQb = j=7- - C, Ta . 



2gJ(\ - P " ) 

 Differentiating and eliminating V, gives as their slope 



9Qv__ iLnl 1 Q + CpTa 

 QP ~ K P <:z} 



P < -1 



The slope of the E curves has been found (note on p. 643) to be 



— — Q 

 1 p " " * 



QQb _ k — 1 Q Cp Tg 



~9P ~^r p «Ei 1^ * 



lege P " -1 + P " 



The necessary and sufficient condition for a point of tangency is %^ = ^r, > 



c)P Qr 

 and this can be put in the form 



K — 1 1— K 



-(log.P " -1-i-P < )=0. 

 that is Fi (P) x2 - F. (P) x - F^ (P) = 0. 



where x = • This is the equation of the desired locus. Each of the func- 



^p ■!■ a 



tions F is positive for values of P greater than 1, so that, of the two real roots of 

 the equation, one is always less tlian zero (by Descartes' rule of signs). The other 

 root gives the line plotted in Figure 13. 



