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



we do not so well know, but there are some who sj)eak of a relatively mild 

 postglacial climate, which the underground temperatures indicate, as we 

 shall see. 



If, then, we assume that when the ice had melted, the temperature 

 rose rather suddenly to the temperature of the bottom of Lake Superior 

 (the temperature of the maximum density of water), and that Lake 

 Duluth was like Lake Superior in that respect, we shall make a reason- 

 able assumption. Yet it is possible that the ice which once filled the 

 kettle-holes may have taken a hundred years after the recession of the 

 ice-sheet to melt. If we assume that since the land emerged above Lake 

 Duluth the mean temperature did not drop, but remained somewhere 

 between the temperature of the maximum density of water (38 degrees 

 Fahrenheit) and the present temperature or higher, but not lower, we 

 make also a probable assumption. There was no readvance of the ice- 

 sheet front, so far as I learn, after the abolition of Lake Duluth. 



ISTow the flow of heat waves into the ground has been a subject of in- 

 vestigation many times,^^ and if we can give definite values for the varia- 

 tion of surface temperature, we can calculate the underground tempera- 

 tures. This we can not do exactly, but we can derive an approximation 

 which will answer present purposes and can be handled by any geologist 

 so as to adapt it to various data without imdue labor in getting approxi- 

 mate results, which will be as accurate as present facts warrant. 



Suppose that the Ice Age lasted long enough so that the temperature 

 gradient was for great depth and below the depths of observation ad- 

 justed to the surface temperature of freezing (32 degrees Fahrenheit 

 under the ice-sheet), and that at the end the temperature rose to 42 de- 

 grees and stayed there. This rise would make an addition to the previous 

 temperatures in the shape of a logarithmic curve of the probability inte- 

 gral. It is the curve figured in figure 1. The maximum ordinate is 

 determined by the rise in temperature. In tlie case chosen this is ]0 

 degrees. 



If one wishes to apply it to some other change in surface temperature, 

 one increases the ordinates, or, what is the same thing, changes the unit 

 of the scale of ordinates in proportion. 



The ratio of the increase of temperature for various depths is given 

 by the other ordinates to the surface increase. 



The depths are connected with the abscissas in this way, that the 

 abscissas are proportional to ni where /» = x/2a \ t, — that is, to the 



" See references in my report for 1903, p. 205 ; also Ingersoll and Zobel. Mathemat- 

 ical theory of heat conduction, chap, vii ; also E. D. Williamson and L. H. Adams, 

 Physical Review, n. s.. vol. xiv. August, 1919, p. 100. 



