MATHEMATICAL DISCUSSION 717 



Fahrenheit and B was 1/90, then the last part of equation (9) becomes 



e (A+Bix+m^/xUd^ 



- / e ( A+B i-x-i-m^/x) W' 



We see that the A terms are equivalent to one-half of Poo Pm — 

 The B terms, taking out B, m, and x from under the integral sign, are 



^'(^— "->.^(/:7:)('"'-* 



The last two integrals are easily solved and are zero, for we may take 

 jS^ as the variable, and d^"^ is 2/3(i/3, and the limits of integration for jS^ 

 are the same, whether /3 is m or — m. 



Also Px = 1 ; thus all the Bx terms reduce simply to Bx, and 



(12) u" = AP^^Bx. 



If i = m=oo and Pm = 1 and u'\xo)A-\-Bx, as it should. 



If, now, we assume for F its simplest value and assume that 



(13) Fit) is simply C, a constant, say the present mean annual 

 temperature of Calumet, (43 degrees) F{t) = C, then the first part of 

 equation (9) becomes: 



(14) u' = C{P„-P^) 



Combining equations 14 and 12, we have this result, that if U(^ot) = C 

 and U(xo) = A-\-Bx 



(15) then Ui,t) = C'+{A-C)Pm-\-Bx. 



It is this formula which we have used to estimate the time since the 

 last Ice Age, but it is only a first approximation. The expression for 

 the rate of increase downward under the ice may, perhaps, be nearly 

 enough covered by the expression A -{-Bx, if the ice ages are reasonably 

 long; but the expression for the surface temperature is surely not uniform. 

 It might naturally be a periodic one with a constant temperature during 

 the times that the region is covered with ice. A simple form on which 

 I have done some nmnerical work assumes u = 0°, and. starts at the 



middle of an Ice Age, from ^ = to t= T/A, then lets u = sin z^—z^ up to 



ST/4:, and then =0 to T, where T is one complete period of climatic 

 fluctuation. 



Such a curve might not be very far from representing the real fluc- 

 tuation of temperature, except that the duration of the ice-cap would 

 vary, growing less toward the margin of the ice-cap and also having 



