MATHEMATICAL SUMMARY 403 



y = g = Cx' where C= KIch' (2 ulu, — 1)"^ 

 y = g" = Ax' -^ B = - 



ch {u 





kc , ^^ kc 



y = g'" =z E=Kln yujiCo = -r= where K = 



a'l/it ay^Uo 



If from an observed curve of the grain we can determuie these tan- 

 gents approximately and the points of intersection, we can find a num- 

 ber of data as to initial conditions by the following formulae : 



JL =f = ElAb hAc 2/^ = 1+ (E/A5 h'Ccf (2 PY 



= 1 -f {El Ah KCcyil + (^/.45 hCcf (1 + 4.cf\ ' 



K= T. Ef= Ach (u/a,)^ ]/¥ = CcK {-^ — l)* 



^'n = -^ = ch' (2 ulu, — 1)"5' l-\/ulu, and is < .282. 

 x',, = (E—B)/A = Abhcr — y' and ^''' "^ '^' is < .4 



x',, = BKC-A) = yl^(l--\-) 

 Zfi \ 2 unto J 



— is between and 



hi/2 _i 

 h' 



We also give as useful tables in this connection a table of the proba- 

 bility integral derived from Johnson : 



u/Uo =^ P^ = -^^ I e '"'•^ dm. 

 y tl J o 



We also give a table giving ul Uo for various values of q. With these 

 in hand, it is not difficult to solve questions that may arise, if the neces- 

 sary facts have been observed. 



The observations to be made in the field, if possible, are as follows : 



The size of the sheet or dike ; 



Indications as to whether the contact zone is broad or narrow and 

 more or less altered ; 



