RELATION OF CAPACITY TO SLOPE. 



97 



directly and approximately in simple ratio, 

 with Q, and (3) that they vary increasingly, if 

 at all, but very slightly with w, while (4) a rela- 

 tion to H could not be disentangled from the 

 relation to F 2 . The discussion of the values of 

 n showed (1 ) that they vary inversely and irreg- 

 ularly with F 2 , (2) that they vary inversely 

 and more regularly with Q, and (3) that the 

 variation in respect to w is direct for small 

 widths and inverse for large widths, while (4) 

 the relation to H is not separable. 



The data on the three parameters are sum- 

 marized in equation (27), which is an expan- 

 sion of equation (10). In all probability /, and 

 f u are as complex as/, but no factors are intro- 

 duced of which the influence was not definitely 

 shown by the discussion. 



and have purposely omitted details. The 

 results, despite important qualifications, show 

 clearly that any general expression of the law 

 connecting capacity and slope which might be 

 based on formula (10) would be highly complex. 

 With reference to the main subject of this 

 chapter, the following section is of the nature 

 of a digression, its purpose being to define a 

 method and terminology used in several of the 

 succeeding chapters. 



THE POWER FUNCTION AND THE INDEX OF 

 RELATIVE VARIATION. 



One of the algebraic forms to which the title 

 power function is applied is 



y = ax n ........... --(28) 



,*, _(27) If the coefficient be suppressed, leaving 



This equation is subject to a qualification 

 connected with the assignment of values to a. 

 It will be recalled that that assignment was 

 somewhat arbitrary, and also that the values 

 of a entered into the computation of the values 

 of &, and 7i. A systematic error in the values 

 of a would therefore cause systematic errors 

 in the other parameters and might vitiate 

 conclusions as to the laws of their variation. 

 A search was made for evidence of such errors, 

 the search making use of the principle (easily 

 demonstrated) that a positive error in a would 

 cause a positive error in J t and a negative error 

 in 7i. While the result of the search was nega- 

 tive, it is not to be supposed that the values of 

 a have high precision. To their errors, in 

 combination with the obscure influences of the 

 varying range of fineness and with the errors of 

 observation, are to be ascribed the irregulari- 

 ties of the constants of the adjusting equations. 



While the algebraic relations are such that 

 minor errors in the values of a might have im- 

 portant influence on values of b t and n, their 

 influence on the interpolated values of C would 

 be small. 



The uncertainties affecting the several ele- 

 ments of equation (27) are so great that no 

 attempt will be made to develop from it a 

 definite and quantitative expression for the 

 relation of capacity to slope. For this reason 

 the preceding paragraphs have attempted to 

 present only the general tenor of the discussion 

 20021 No. SO 14 - 7 



y oc x n 



this is the exact equivalent of the familar 

 "y varies as the Tith power of x." This mode of 

 comparing the rate of variation of one thing 

 with the rate of variation of another is exten- 

 sively employed, and it so commends itself by 

 its simplicity that its use is constantly extended 

 into fields where its applicability is approxi- 

 mate only. Having occasion to make much use 

 of certain variants of this function, I find it 

 important to obtain a clear conception of its 

 properties and shall therefore give the matter 

 somewhat elementary attention with due 

 apology to the mathematical reader. 



If we consider x and y merely as numbers, 

 the rate of variation with y with respect to x 

 is the ratio of the differential increment of y 

 to that of x. That ratio is 



.(29) 



If we consider x and y (and also the constant 

 a) as powers of a common base, equation (28) 

 becomes 



log y = log a + n log x (30) 



The rate of variation of log y with respect to 

 log x is, differentiating, 



-(31) 



dlogx 



