THE PERCENTAGE COMPOSITIONS OF ROCKS 283 



a vertical edge (Fig. 6) forming a vertical prism whose dimensions 

 are d, d, and D; the value of d, of course, differing from the dj 

 value but such that Ddi = d^. The surface and volume values 

 in each case, naturally, are the same. This may easily be seen 

 by considering a solid built up of black and white cubes. 



I. CASE OF CUBICAL GRAINS, UNIFORMLY DISTRIBUTED 



a) Assume a cube, 10 inches to a side, built up of small black 

 and white inch cubes, uniformly distributed. If the black-white 

 ratio is Sc»-5o, every other block must be 

 black. Moved to one side so that all the 

 black cubes are together (Fig. 5), the num- 

 ber at the surface will be 50, distributed in 

 a rectangle whose sides are d^, and D (5 Xio). 

 Moved to one edge the 50 square inches of 

 surface would appear as a square, geven 

 cubes on a side and one left over; in other ^^^- ^ 



words, (^ = 7.07 + . The total number of black 

 cubes is 500; therefore the volume is not represented by d^ 

 ( = 353.5+) but by ^^Z) = 50X10 = 500. The relationship, there- 

 fore, between the black minerals appearing at the surface to the 



entire surface is as jr^ , while the volume relation of black to white 



. Dd' , d' ^ d' Dd' . . r . 



is -T^rr and not i=r, . But ^r, and -prr are equal; therefore the ratios 



£)i £)o D^ D^ -17 



of surface and volume measurements are equal. 



h) If the ratio of black to white is very small, such as i to 

 10,000, it will not do to assume that the black mineral occurs as 

 a single cube, whose volume is ^V-cubic inch, in a lo-inch cube of 

 white, for that would vitiate the original assumption of uniform 

 distribution of constituents. If we say that the ratio of surface to 

 solid is as d"" to d^ it means that our mineral all Hes in the top layer 

 and is not uniformly distributed. It is comparable to saying that 

 a topaz granite in which an accidental crystal of topaz occupies 

 half the slide is half-topaz, while as a matter of fact there might 

 be but a single crystal to a cubic foot. Consider, therefore, that 

 the small amount of black mineral is uniformly distributed in 



