KELATION OF CAPACITY TO SLOPE. 



123 



straight line; and the drawing of the line gave 

 values of B and n a in E = B (S-aJ"". In 

 each of the remaining five cases the indicated 

 locus was a curve, and the curvature was such 

 as to indicate a larger constant in place of a. 



This larger constant, a 1} was determined graph- 

 ically, and the other parameters were com- 

 puted as before. The results are given in 

 Table 23a, together with comparative data 

 from Table 15. 



TABLE 23a. Comparison of parameters in the associated functions of capacity and efficiency, C=6,(S a)n and 



E=B (S a,)"". 



The proximate inferences from these plots 

 and comparisons are, first, that efficiency may 

 be formulated, with sufficient accuracy for 

 practical purposes, as proportional to a power 

 of S a 1 ; second, that, when it is thus formu- 

 lated, the approximate values of a l are in gen- 

 eral larger than the values of a obtained in the 

 formulation of capacity; and, third, that the 

 values of the exponent are smaller than the 

 equivalent values for capacity, the differences 

 usually being somewhat less than unity. 



The field of these inferences was also tra- 

 versed by a mathematical inquiry, of which 

 the results are more definite. If the relation 

 of efficiency to slope be formulated by 



E=B(S-o l ) n 



-(47a) 



the exponent is always less than n, but never 



so small as n 1. For the range of conditions 

 covered by the experiments, it is little greater 

 than n 1 . The value of a t is always greater 

 than the corresponding value of a, the differ- 

 ence being usually small. The difference is 

 greater when the value of the exponent is 

 relatively small. Equation (47a) is incom- 

 patible with the corresponding equation for 

 capacity, (10). If the locus of E=f(S) be 

 separately plotted by means of the two equa- 

 tions, the resulting curves are not coincident, 

 but they intersect at three points and lie close 

 together elsewhere (in the practical field) 

 unless the difference between a and a l is large. 

 On the whole, it appears entirely feasible 

 to formulate efficiency by means of equation 

 (47a). 



