﻿200 DE. H. C. SOKBY ON THE APPLICATIOJiT OP [May I908, 



than the final velocity, it seems probable that for granules '001 

 inch in diameter the velocity of the current could not be above 

 •01 foot per second, equal to about 36 feet per hour, and for those 

 •0001 inch in diameter about 3| feet per hour or less. At all 

 events, for fine-grained clays these velocities are probably of the 

 true order of magnitude, though possibly too great. 



XII. On the iNTEESrACES BETWEEN THE CONSTITUENT GeAINS 



OF Deposited Material. 



A knowledge of the relative volume between the constituent 

 grains of rocks, as originally deposited, or as modified by subse- 

 quent mechanical or chemical changes, throws much light on many 

 interesting questions. In studying this subject the foundation is 

 to a large extent mathematical ; and, in order to facilitate calcu- 

 lation, it was desirable to assume that the grains are spheres of 

 equal size, uniformly arranged in various ways, so as to occupy as 

 much or as little space as possible or some intermediate amount. 

 Possibly this problem has never before been treated from its geolo- 

 gical side. For the sake of simplicity, I have made my calculations 

 as though there were only eight spheres, but so treated the question 

 that the results would be the same as if the numbers were so great 

 that the effects of the outer surfaces coiild be neglected. 



(1) The first case to consider is when four spheres are arranged 



as a square, and the other four placed directly over them, so that 



each sphere rests upon only one, and the bounding surface of the 



whole is a cube. The radius of each sphere is taken as unity, 



and therefore the length of each side of this cube is 4, and the 



volume 64. The united volume of the spheres themselves is then 



4 . 33*51 



-7rx8 = 33'51. Hence their relative volume is ———= 52*36 per 



cent., and of the interspaces 47' 64 per cent. 



(2) The next case is when four spheres are arranged as a 



square, and the other four tilted over, so that each rests upon two, 



and the bounding surface is a parallelepiped, four sides of which 



are squares and the others parallelograms having angles of 60° 



and 120°, so that the height above the square base is 2V3, and 



the volume 2 \/3 x 4 x 4 = 55*42, whereas that of the spheres alone 



33 '51 

 is 33-51. Hence the relative volume is -^^-77^= 60*46 per cent., 



and the relative volume of the interspaces is 39*54 per cent. 



(3) The third case is when four spheres are arranged as a square, 

 and the other four tilted over in the line of one diagonal, so that, 

 if the number were indefinitely great, each would rest upon four. 

 This would give a parallelepiped having a square base, and two edges 

 inclined at 45° to the base, so that the height would be 2 V2, and 



the volume 2\/2x 4 x 4 = 45*25. Hence the relative volume 



33'51 

 would be -^^25 ~ 74*05 per cent., and that of the interspaces 25*95 



per cent. 



