718 A. C. LAXE GEOTHERMS OF LAKE SUPERIOR COPPER COUNTRY 



more or less lag after the climatic minimmn in extending and after 

 the climatic maximum in retreating. 



Such a curve may be represented by a series of sine tenns 



(16) u' = F{t) = - ^— +- (sum of terms (-1) [^^^^ZY)) 



sin 2Trnt/T, and if we represent F{t) b}^ any such series of sine terms 

 sin {2Trnt / T -\-l) we can get a general expression for u' (xt) by writing 

 for each term e — D (sin 2nt-\-l—D), where D = {x/ax\/ir/T as above. 



The resulting series are, however, not always convergent or only 

 semi-convergent, though they are the more rapidly convergent the 

 greater x and the less t, and I have not carried the mathematical work 

 numerically farther, as we know so little regarding post-glacial climate 

 fluctuations. It may be noted however, that each sine term represents 

 an effect on the temperature which is later, and less, the deeper the 

 point the temperature of which is studied. 



If we take a relativel}^ simple assumption, we can get a closer approxi- 

 mation to the actual readings. Assume that there was not merely a 

 sudden jump to a temperature of C( = 43 degrees), but that there is a 

 simple sine variation of temperature added. This will bring in three 

 new constants which we can juggle with, to wit, the amplitude of this 

 variation (K), the period of duration of it (T), and the time at which it is 

 zero (L). It is therefore no wonder if we can, by giving them certain 

 values, get prett}^ good accord. If we note that the actualh^ observed 

 temperatures are most in excess of those obtained b}'^ the equation 



(15) Uuo = C + (A-C)P«+J5x = 43-i-(32-43)Px/2V2r3^+l/90a; for 

 about 1,8C0 feet depth, where they are about 8 degrees in excess, we can 

 get a good approximation by assuming that the crest of the periodic 

 wave above mentioned is now at that depth, and that it was at the 

 surface when the ice went off. Whence we get 



(16) 27r(ll,080) T 1800vV2.3r +L=7r/2 



(17) 27r(0)/T = 0\/7r 2.3r +L = 7r/2 



Eliminating from these equations, we find that this would involve a 

 period T of about 90,000 years. 

 Then by V 



(18) Ke ~ lS00V7r;2J3(T=90,0()C) _ g _ ^^ - 224 /Vr 



we find that K= about 17 degrees if this is the value of T. 



This would imply an immediately postglacial mean temperature of 



