402 A. C. LAXE — PORPHYRITIC APPEARAXX'E OF ROCKS 



might obviously be veiy mucli more conspicuous and the same time 

 automorphic near the margin. Suppose, for instance, that in a granite 

 the initial temperature was such that Uol2 was within the range of forma- 

 tion of the feldspar, while the quartz formation was at a considerably 

 lower temperature. The feldspar would be conspicuously coarser near 

 the margin, while the quartz might not vary much from center to mar- 

 gin, having a curve or grain something like that numbered .40 of plate 58. 



Mathematical Summary 



We may briefly summarize the results of the mathematical investiga- 

 tions as follows : 



In an injected sheet of uniform temperature and conductivit}' whose 

 walls are kept at a fixed temperature the temperature may be expressed 

 in one of two ways, the former most applicable at the earl}^ stages of 

 cooling, the latter in the latter stages of cooling. 



— = P—(P —P ]-{-(P —P )—..,, 



—. = q sin ~ x/ c -\- -jr- sin S~ xj c -f -^ sin 5 - xl c. 



4 Uo O D 



(rr I m )^ 



Where m = xl2a\,'' t and 7?7o = c/2ai/ t and (/ = £ 

 and (see page 406) P^ = — = | e~'^ dm. 



i/tz J o 



From these we may derive two formulae for the grain approximately, 

 one of which represents a tangent to tlie grain at the margin, if we as- 

 sume a certain part of it — namely, h not to depend on x. We give above 

 a table for h. The other represents the grain at the middle and a tan. 

 gent to the grain at the middle, provided the consolidation takes place 

 in the latter stage of cooling. In case we do not assume that the walls 

 are kept at a fixed temperature, we ma}^ find the temperature as func- 

 tions of those of a dike whose walls are kept at a fixed temperature by 

 the following formula : 





And for the grain we can derive a formula which is applicable at the 

 margin, another applicable at the center, and another applicable between 

 in case the contact zone is not large, which will represent three tangents 

 to the real curve of the grain as follows : 



