716 A. C. LANE GEOTHERMS OF LAKE SUPERIOR COPPER COUNTRY 



Let W(j<) be the temperature at a depth of (x) at a time represented 

 by (t), and let 



(1) u^xt) = u\xt) + u'\xt), where 



(2) u\^^ is such that u\xo)= 0, and u\ot) = F{t) — that is, is a function 

 of the time only, at the surface, where x is 0. 



Also: 



(3) u'\xt) is such that u'\ot) = 0, and u'\xo) =f(x). 



(4) Then u^xt) will be = F(t) at the surface — that is, 



u^ot) = F{t). 



(5) Also 



U(^XO) = / (^)- 



But u must also satisfy the differential equation 



(6) Dtu = 0} D\ u. 



A value of u^\xt) is found in Article 50 of Byerly and of u\xt) that will 

 satisfy this equation (6) as well as the conditions above (by Byerly equa- 

 tions (6) and (7) of page 84, and (10) of page 88. Compare Ingersoll 

 and Zobel, page 70, equation (22),) and we have: 



2 f^ 'f^ 2^2 



(^) ^-=V^i./2aV7^ F(^-.V4a ).d^ 



and if we replace the second part by the value given in Byerly's equa- 

 tion (7), page 84 and also write 



(8) m = x/2a;\/'t, we have : 



(9) u^xt) = -^\ € ■F{t-mH/^'')d^-\—j. e ■f(x+m^/x)d^ 



'« — /32 



6 f{-x-\-m^/x)d^. 



m 



If the functions F and / are known, these expressions may be expanded 

 into strings of probability integrals of the form 



Pm = -j^ / e dm. 



V 7r7 



Suppose, for illustration, 



(10) f{x)=u'\xo)=Uixo)=A-\-Bx, 



where we may assume that under the ice-sheet A was about 32 degrees 



