212 



Capt. M. Greenwood. 



hence, up to W = k, 



1 = <^H- 1 d/(W) _ dcf>iW) 

 E ^^W E' ^^W f/W 



. h, 



and consequently E might differ considerably from E', for values of W not 

 exceeding k. But thereafter it will become constant and equal to E', for 

 d/(W)/dW and d(l)(W)/dW will vanish. 



The argument may be illustrated on Benedict and Cathcart's cyclist 

 working at rates of 68-72 revolutions per minute. Here, for W = 0"48 or 

 greater, H = 3'3415 W + 2"4131. Let us assume that constancy is reached at 

 W = 0-45, which will be the k of our formula. Hence E' = 0-29926 ; 

 putting h = 1"17 (its mean value in Benedict and Cathcart's series), we 

 have : — 



We have no knowledge at all of (/>(W) and /(W) save the initial conditions 

 of each and the terminal value of their sum. Assuming that (f> (0'45) = 1, 

 and that each is linear for that range, we should have 



(f>(W) increasing by 0'222 for each 01 calorie for to 0'45 and 



/(W) increasing by 0-836. 



Hence for values of W up to 0'45, E = 0"1639 ; for greater values of W it is 

 equal to 0'2993. The locus of H is formed by two straight hues intersecting 

 at W = 0'45. Naturally the postulated forms of ^(W) and/(W) are merely 

 illustrative of the way in which the ultimate approximately linear relation 

 might arise. Generally, the hypothesis is that 



subject to F{0) = h and F (oc ) = and the practical condition that 

 d¥(W)/dW shall be very small for W greater than a constant value k, say. 



There is some formal analogy between this expression and the well-known 

 equation of Chauveau,* and the point arises as to the sense to be attributed 

 to the value W = 0. 



It is evident that the proposed equation is not even now complete, because 

 the assumption of E' = a constant ignores the non-compensated element of a 

 real thermodynamic transformation (i.e., if in such an incompletely reversible 

 cycle, H' is the heat of the transformation, and T the absolute temperature, 



is not zero). It is, however, convenient to examine the case on the 



assumption that W = corresponds to the point where no external work is 



1-17 [!-</> (0-45)] -|-/(0-45) = 2-4131. 



H = W/E' + F(W), 



* See Lefevre, op. cit., p. 728. 



