EXPERIMENTS WITH MIXED GRADES. 



183 



of debris is measured. It may be a unit of 

 weight or a unit of volume. In this report a 

 body of debris is conceived to be composed of 

 equal volume units, each of which has a deter- 

 mined or determinable fineness, and its mean 

 fineness is the mean of the finenesses of the 

 volume units. In dealing with bulk fineness 

 the mean computed is the arithmetical mean 

 of the finenesses of units. In dealing with 

 linear fineness the mean computed is the cube 

 root of the arithmetical mean of the cubes of 

 the finenesses of units. 



The intricacy of the definition of mean linear 

 fineness arises from the relation of linear fine- 

 ness to bulk fineness. The fundamental con- 

 cept is that of bulk fineness, and the definition 

 of linear fineness rests upon it. Linear fineness 

 is essentially a derivative of bulk fineness, and 

 mean linear fineness is an exactly similar de- 

 rivative of mean bulk fineness. To pass from 

 an assemblage of linear finenesses to their 

 mean, it is necessary to pass through bulk 

 fineness, and that passage involves cubes and 

 cube root. 



Bulk fineness is defined as the number of 

 particles in a unit volume (1 cubic foot), it 

 being assumed there are no voids. It is the 

 reciprocal of the volume of the particle which 

 might be called bulk coarseness. There are 

 two practical modes of measuring it. If the 

 specific gravity of the debris be known (or 

 assumed), measurement includes a weighing 

 and a counting. Then, W being the weight, 

 N the number of particles, G the specific 

 gravity, TP the weight of a cubic foot of 

 water, and F t the bulk fineness, 



W.GN 



~~ 



.(86) 



If the specific gravity be not known, measure- 

 ment includes two weighings and a counting. 

 Then, W being' the weight in air and W, the 

 weight in water, 



.(87) 



This procedure determines bulk fineness 

 when all particles have the same volume; when 

 they are of different volumes it determines 

 mean bulk fineness. As a matter of fact, all 

 our measurements in the laboratory were of 

 mean fineness. It is not possible by any 



method of sorting with which I am acquainted 

 to separate from a natural alluvium a grade 

 which is really uniform in fineness. 



When the mean fineness of a sample of 

 debris is desired, there is no need to separate 

 it into grades, because the process for measur- 

 ing the mean fineness of the whole is identical 

 with that for measuring the fineness of a grade, 

 When bodies of d6bris of known finenesses are 

 mingled, the mean fineness of the mixture is 

 computed by a formula (85), which sums the 

 finenesses by unit volumes (or weights) and 

 then divides by the number of unit volumes. 



Linear fineness is defined as the reciprocal 

 of the mean diameter of the particles of the 

 d6bris. Like bulk fineness, it is treated as a 

 property of the body of debris and not as a 

 property of the particle. Mean diameter is 

 defined as the diameter of a sphere having the 

 same volume as the particle. Defined thus, 

 linear fineness is a function of volume of parti- 

 cle, and as bulk fineness is also a function of 

 that volume, the two have a fixed relation: 



F-(5 



.(88) 



Substituting in (88) from (86) and (87), we 

 have 



The computations of fineness for this report 

 used (89) or (90) ; or, what is equivalent, they 

 first determined bulk fineness by (86) or (87), 

 and then derived linear fineness by (88). The 

 computations of mean linear fineness applied 

 (88) to mean bulk fineness. 



It would have been possible to formulate fine- 

 ness in such a way that the definition of linear 

 fineness would be direct and comparatively 

 simple, but any such formulation would en- 

 counter complexity in some of its parts, pro- 

 vided it established a logical relation between 

 linear fineness and bulk fineness. Its adoption 

 would also -involve the sacrifice of simplicity in 

 the measurement of fineness. Any system re- 

 quiring the direct measurement of diameters 

 would be inferior for practical purposes to the 

 one here used. 



The subject of scales of fineness has been 

 elaborated because nearly all the results as to 



