284 ALBERT JOHANNSEN 



cubes measuring yV i^^ch on a side (^=tV inch). If now all these 



small cubes be gathered along one of the vertical edges of the 



large cube, the portion of the black mineral of the entire block 



which will appear at the surface (Fig. 6) will be d^ = (tV)^> and 



d" (rV)^ 

 the black-white surface ratio will be Yi2~ 2 ^To-o-irir- That is, 



the black mineral forms .01 per cent of the surface. The volume 



Dd^ (tV)^ 



"^ = '°^ 

 ratio, therefore, is the same as the surface ratio. 



ratio of black to white is ~^;:;- = io — o"=T¥T¥ir- The volume 



n. CASE OF TABULAR PLATES, UNIFORMLY DISTRIBUTED 



Let a, h, and c represent the ratios of length, breadth, and 



thickness of tabular dark constituents. If they are uniformly 



distributed the same number of flakes will be contained in each 



layer below the upper surface; consequently no matter what the 



thickness, Dah represents the volume of the black blocks, while 



D^ represents the large block. The ratio of surface measure- 



. ^ . ab . ^ - Dab ab , 



ments to the whole is y^, and of volumes —^ ^Tu'^ consequently 



the two are equal. 



With thinner flakes there would be more layers in the large 

 cube, and with thicker, less. But no matter what the ratio of 

 a to 6 to c, the surface percentage is the same as the volume per- 

 centage, provided the grains are equally distributed. Of course a 

 laminated rock, with layer of different minerals, might be so cut 

 that one slide would contain no dark minerals and one no light. 

 A cross-section of such a slide, however, would give correct results, 

 for the distribution at right angles to the slide is uniform. 



in. CASE OF CLOSELY PACKED CYLINDRICAL RODS 



This case is approximately the same as the preceding, for in 



one direction in the slide the mineral is continuous while in the 



other there is interstitial material. Sections should, therefore, be 



taken at right angles to the elongation. The surface ratio is 



irr^ Divr^ 



777 , and the volume —fr- . As before, the two are the same. 



