APPROXIMATE FORMULAE 397 



the cube root varies but slowly, it will be easy to insert an approximate 

 value in (16) and obtain nearer approximations. Moreover, having 

 found ulu^ we can go on to find K. 



(17) K=7z eJjL = Cch' (2 u/Uo — 1)* = Ach (w/t^* i/2~. 



An interesting question is where the various tangents meet — that is, 

 where will the zone of marginal grain become equal to that of the center. 

 If we refer to g^ and g"\ we have the following formulse : 



(18) x\, = Elc = ch'(2r-iy/7rf, 



where a/jg is the abscissa of the meeting point and x\^/c is less than .282 ; 

 and if to g'^ and g''\ similarly 



(19) x',, = (iE—B)IA = Ad hcf — y' , 



where {x^.^.^ + 2/0/^ is less than .4 ; and finally lines g' and g'\ formulae 10, 

 11, 12, will meet at a point given by 



(20) .',,= 5/(C-^) = W^(l-2^.)"'-l. 



From formulae (18) and (19) we shall not be able to find the width of 

 the contact zone, but we may often be able to if we know over what 

 range some of these formulse are closely applicable. If, for instance, 

 formula 10 holds at least to a value a;', it may be shown that 



(21) %) is not less than a;VP~" ' (2 w/tto) — 1 or 2w. 



3. 



Also (in equation 20), if we know x^,^ y = ( ^iyTf — — ^V^" 



= (l.4toc.J,-l).. 



EOCRYSTALS OR PhENOCRYSTS FORMED SOON AFTER ReST 



An interesting application of the view point which we have developed 

 may be made to the granite dike some 2,000 feet wide cut by the Wachu- 

 set aqueduct and exposed on Carvill hill, near Clinton, Massachusetts, 

 which has been very carefully studied by W. O. Crosby,* who has kindly 

 guided me over the ground, so that while I quote from him I am also 

 speaking from personal observation. 



*TechnoIoa;y Quarterly, vol. xii, no. 2, June, 1899, pp. 68 to 98. 



