THIRD HYPOTHESIS OF DERIVATION 461 



mined by the geometric process, or a by photographiiio- or by drawing ; 

 but in the calculation little accuracy can be expected from dependence 

 on use of either d, d\ or a in place of cP. 



Third, .7854|/'(i^ or a. The prevalence or predominance of grains im- 

 perfectly filled out, and thus lacking cubical dimensions, in certain 

 rocks — for example, many subcrystallized igneous and metamorphic 

 varieties, rocks with spherulitic texture and conglomerates — indicates 

 conditions of approach to spherical form and the propriety of use in 

 calculation of the last formula to determine the proportional relation- 

 ship of the true spherical volumes of such plates in the thin-section. The 

 factor d may be estimated by the geometrical process, or a by the photo- 

 graphic process or by the drawing process of Sollas, which are described 

 beyond. With the massive rocks, to which the foregoing three methods 

 are applicable, the thin-section may be sliced in any plane. 



Fourth, d^ (= a). In many foliated rocks this equation holds true, 

 those whose grains are mostly isodiametrical on the schist plane, with 

 uniform and less thickness, particularly thinly laminated rocks, most of 

 whose elements are thin plates. In such, d may be determined on that 

 plane by the geometrical process and then used as d^ in the calculation 

 of proportional volumes, or a determined directly by the photographic 

 process or drawing process. 



Fifth, a\ We have yet to consider those foliated rocks or schists in 

 whose predominating grains one dimension much exceeds the others — 

 those made up of elongated scales, blades, prisms, needles, and fibers. 

 From the ease of splitting off a chip in such laminated rocks, their thin- 

 sections are usually prepared from a flake so obtained. In these, how- 

 ever, the maximum areas of the elongated plates are presented, which 

 would possess an exaggerated value in calculation of proportional vol- 

 umes of the minerals. This fact is partly recognized by Rosiwal, who 

 recommends : 



"Fiir Schiefergesteine von hoher Parallelstructur geniigt ein Biindel weniger 

 paralleler in einer zur Schieferung annahernd senkrecht stehenden Schliffebene 

 gelegener Mengenlinien," * et cetera. 



His modification, however, is unnecessary in the simpler variety of 

 foliation provided under the fourth condition, and in the remaining 

 cases insufficient, in that it disregards the additional orientation of such 

 a section, " approximatel}^ perpendicular to the schistosity,'' needed in 

 reference to the direction of fibration and other conditions, in schists 

 made up of parallel elongated elements. In these, three cases are to be 



*Loc. cit., p. 148. 

 LXIV— Bull. Geol. Soc. Am., Vol. 14, 1902 



