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



Then plot from the same origin a curve which is the reverse of the 

 probability curve — that is, through the following points '^- 



X = .000 



.OS 



.18 



.36 



.78 1.00 1.10 1.20 1.40 1.54 1.64 1.84 



y = 1 



.91 



.80 



.61 



.27 .157 .12 .08 .05 .03 .02 .01 



These values of ij represent the eifect of a sudden increase of tempera- 

 ture at a time past (/) and at a depth such that if or is the diffusivity 

 (= 203 here, 400 (Kelvin) for units of foot and 3^ear) the abscissa m = 

 (depth 7/)/2aVr 



Xow expand this probability curve reversed up, if necessary, by in- 

 creasing every ordinate in the same jDroportion so that it shall be close to 

 the surface temperature observations at the surface. Then see how much 

 we can stretch it to the right, increasing all the abscissas in temperatures 

 the same ratio, and still have the observed excess lie above the curve. If 

 we let m = 1 correspond to 4,000 feet, most of the observations will lie 

 l)ek)w the curve in figure 1, and if m = 1 corresponds to 2,000 feet, the 

 curve lies way under the o])servations. The nearest fit is by letting 

 in = 1 correspond to 3,000 feet and stretching the curve to the right 

 about three times. Whence, using 203 for the diffusivity instead of Lord 

 Kelvin's 400, as it agrees better with B. 0. Peirce's tests on these very 



rocks, we have 1 = 3,000/(2 X 14.4 xV ^T- Whence we have / =^ 

 11,080 years since the rise started to spread downward. This checks well 

 enough with recent estimates by de Geer and Antevs. 



It is clear that any other diffusivities or depths at which postglacial 

 changes of climate are not yet perceptible can be readily placed in the 

 above simple equation, which may be expressed in words as follows : 



The depth at which the excess of temperature, above that given by the 

 gradient derived from temperatures too deep to be affected by postglacial 

 warming for the said depth, is but 15.7 per cent of the excess of the sur- 

 face temperature, is twice the square root of the diffusivity multiplied by 

 the time since the sudden postglacial amelioration. 



Lord Kelvin took the diffusivity cr as :^ 400. Other figures are given 

 in Ingersoll and Zobel's Appendix A, but they omit B. 0. Peirce's figures 

 on the copper-bearing rocks,^^ to wit : 



12 Davis, Brenke, and Hedrick's Calculus Tables, p. .54: .Tohnson's Theory of Erroi-. 

 aud Ingersoll and Zobel. '•Theory of Heat Conduction." n. 144. * 



" Proceedings of the American Academy of Arts and Sciences. May, 1903. vol. xxxviii. 

 no. 23, pp. 658 and 659. 



The diffusivities in c. g. s. units, used by B. O. Peirce. Callendar aud McLeod. Inger- 

 soll and Zobel, and physicists generally, must be multiplied by 8:^800 to put them into 

 the foot-year units used by Lord Kelvin, and by 3.158.6 to turn them into meter-year 

 units. There is a decimal point wrong in the tigures for meters on page 199 of my 

 report for 1903. 



