On the Efficiency of Muscular Work. 



211 



The argument can best be expressed symbolically. 



Let k be the resting heat production in unit of time, W the work performed 

 (ia thermal units), E' the real efficiency, i.e., W/E, the energy needed to 

 develop W units of work, and H' = H — h. Then we have 



H' = W/E'-0(W)/i 



subject to the conditions 



(i) ({) (W) cannot exceed 1 and ^ (0) = 0, 



(ii) (l-E'){W/E'-<j>(W)h) + h[(l-(l>(W)] = h + \, 



where \ is an unassigned function of W. 



rrv. • ^^w^ (l-E')W/E'-\ 

 These give 4> (W) = ^ ^2_W)h ' 



and <f> (W) attains its greatest possible value when 



W= j-5^{2-E')A+\}. ■ 



until W = {(2-EO/t + X}, 



1 — E 



and thereafter H = 



E 



^^^^^ E = rfW = E'-^ (2-EO 



and subsequently = r^. 



E 



This is, however, inconsistent with the experimental facts, viz., that when 

 W is large, although H is approximately linear in W, it is not equal to aW 

 but to aW + b, and arbitrarily assumes \ to be given by ^(W). Hence we 

 must write our equation in the more general form : — 



H = W/E' + [1-<^(W)]A+/(W) 

 subject to (i) <^(0) = /(O) = 0, 



(ii) W[(l-E')/E'] + [l-</,(W)];i+/(W)< /i, 



(iii) At and after W = k, 



H = W/E' + &, where h is constant. 



From (ii) we have (j> ( W) > '(^^ + 



k E h 



from (iii) A ( 1 - <^ (/, ) ) + /(/.:) = h, 



