﻿Vol. 64.] QUANTITATIVE METHODS TO THE STUDY OF ROCKS. 201 



(4) The last case that we need consider is when the base itself 

 is not square, but the spheres so shifted that each one touches two, 

 and the base is a parallelogram, having angles of 60° and 120° ; and 

 the other spheres are arranged as much as possible in the same 

 manner. It is, however, impossible to have each one resting upon 

 three when the number is indefinitely large, but there are alternate 

 rows with 1 on 3 and 3 on 3 cross ways. We thus get for the axes 

 of the bounding parallelepiped 2 \/4-sec^30°, 2^3, and 4 : so that 

 the volume is 2V'4— sec-30° x 2 V3x 4=45-25, or exactly the 

 same as in the last case considered, and the relative amount of 

 interspaces 25-95 per cent. This is a very interesting result, since 

 it shows that, when occupying the least volume, the spheres could be 

 moved about considerably, without altering the volume. I may say 

 that, in order to test my calculations, I made very careful measured 

 drawings, and obtained practically the same results. 



Experiments with Spherical Shot. 



My experiments were made in a glass bulb holding a known 

 weight of water, and the amount of interspaces was ascertained from 

 the weight of water between the grains, when full of water ; and, 

 in other cases, from the weight of the material used, compared with 

 that of an equal volume, if it had been solid lead. When the 

 small shot was filled' into the bulb without shaking, the volume of 

 the interspaces was 47'2 per cent., which thus agrees closely with 

 47-64 per cent., calculated for spheres occupying as much space as 

 they can when each touches six. When the glass bulb was turned 

 about and well shaken, so as to cause the shot to occupy as small a 

 space as it would, the interspaces were reduced to 40 per cent. 

 This agrees closely with 39-54, found by calculation in case No. 2, 

 for spheres arranged rectangularly in two directions, but one over 

 two in another. It is scarcely probable that such an arrangement 

 is brought about by shaking, but the above agreement is remarkable. 



I then hammered the same shot, thus obtaining disks with a 

 diameter about three times their thickness. On filling these gently 

 into the bulb and afterwards well shaking them, the amount of 

 interspaces was found to be practically the same in both cases as 

 if they had been spheres. This is somewhat remarkable, but of 

 much interest in connexion with sand built up of grains of irregular 

 shape ; for it shows that, if these are of fairly-uniform size, in the 

 long run they occupy nearly the same volume as if they were 

 spheres. 



Experiments with Quartz-Sand. 



As might be expected, the results differ materially with sand of 

 different character. In the case of the somewhat coarse and 

 angular sand of the Millstone Grit, having grains on an average 

 •05 inch in diameter, when filled in variously, but not shaken, the 



