RELATION OF CAPACITY TO FINENESS OF DEBBIS. 



153 



Thus it appears possible that the superior 

 mobility of grade (D) was determined by 

 properties other than size. Unfortunately the 

 record is not of such character that the value 

 of this suggestion can now be tested. 



A third suggestion pertains to the gaging of 

 fineness. The method of gaging included a 

 weighing and involved certain assumptions as 

 to homogeneity in average density and in shape 

 which may not have been fully warranted. 



These various suggestions, while not suscep- 

 tible of test at the present time, are sufficiently 

 plausible to show the possibility of definite 

 causes for the discordances discovered by the 

 comparison of data from different grades. In 

 my judgment it is proper to ascribe the greater 

 discordances to such causes, and to view them 

 as abnormalities with respect to the law con- 

 necting capacity with fineness. 



In view of the magnitude of the abnormalties 

 or discordances, it does not appear profitable 

 to extend the readjustment of data to other 

 and shorter sets. The five sets in Table 44 

 were selected because they included great range 

 in fineness, and because they were qualified to 

 yield fairly definite values of the constant </>. 



VARIATIONS OF THE CONSTANT <f>. 



The laws which control the variation of 

 have not been developed from the observations, 

 but their general character may be inferred 

 deductively by considering the relations of com- 

 petent fineness to various conditions it being 

 assumed that <J> is intimately related to com- 

 petent fineness. Postulate a current of which 

 the velocity is determined by a particular slope, 

 a particular discharge, and a particular width. 

 For this current a certain fineness is competent. 

 Increase of slope or discharge increases the 

 velocity and makes a lower fineness competent. 

 Decrease of width, which corresponds to in- 

 crease of form ratio, increases velocity and 

 makes a lower fineness competent. Thus com- 

 petent fineness, and therefore <f>, varies inversely 

 with the slope, discharge, and form ratio. 



A) ________ --(76) 



capacity and fineness. The formula for the 

 index (cf. pp. 100 and 141) is 



-(78) 



INDEX OF RELATIVE VARIATION. 



Framing an equation of the type of (33) 



C=v i F { < ............. (77) 



in which i t is the index of relative variation for 



With this formula, values of i t were computed 

 from the data in Table 44, and they are given 

 in the lower part of that table. 



By inspection it appears that the index in- 

 creases as fineness diminishes, its growth being 

 at first slow but becoming rapid as competent 

 fineness is approached. Because of the dis- 

 cordances of the data it is not easy to derive a 

 body of values of the index for discussion in 

 relation to other conditions, but it is relatively 

 easy to obtain comparative values of the syn- 

 thetic index, I t , and the variations of these 

 values may be assumed to show the same trends 

 as the variations of i t . Values of 7 4 were com- 

 puted between corresponding data of grades 

 (C) and (G) by the formula 



j log C l - 



-logtf- 



in which (7, and C a are specific capacities corre- 

 sponding to the finenesses F, and F n ; and the 

 results are given in Table 45. 



From these results it is inferred (1) that the 

 index varies inversely with the slope, (2) that 

 it varies inversely with discharge, and (3) that 

 it varies directly with width, and therefore 

 inversely with form ratio. The response is in 

 general of a very pronounced character, but to 

 this there is exception in one of the compari- 

 sons with width. It is possible that the index 

 is a maximum function of width and a mini- 

 mum function of form ratio. With some reser- 

 vation on this point, we may generalize: 



1, P, &)-. -.(79) 



If equation (79) be compared with equations 

 (39) and (68), it will be seen that the variation 

 of the capacity-fineness index observes the same 

 laws of trend as the variations of the capacity- 

 slope index and the capacity-discharge index. 

 In view of this general parallelism of variation, 

 it is thought that the relative magnitudes of 

 average i t , average i,, and average i 3 may be 

 adequately discussed by means of a moderate 

 number of comparisons. Accordingly only 

 those values of i t computed from the five equa- 

 tions of Table 44 are used. The corresponding 

 values of i 3 are not available, but those of i, 



