338 Rev. 0. Fisher on Theories to 



crease per unit depth, and b the mean constant temperature 

 at the surface. For convenience we will take the temperature 

 b as the mean above the freezing-point, which for degrees Fahr. 

 will make b = 20°, supposing the mean temperature of the 

 surface to be 52°. This merely implies shifting the zero of 

 the scale up through 32°. 



Let us then suppose that, as soon as the ice-sheet is formed, 

 the surface is reduced to the freezing temperature of water, 

 and kept at that temperature as long as glacial conditions 

 last. These hypotheses will deviate from the truth in two 

 respects : first, because the mean temperature of the surface 

 will be not suddenly but gradually lowered while the climate 

 becomes colder, before the ice-sheet is permanently established; 

 and, secondly, because the temperature of the surface under 

 the ice will, on account of the pressure, be somewhat lower 

 than the melting-point at atmospheric pressure. Nevertheless 

 our result will give a general idea of the rate and extent of 

 the refrigeration of the crust of the earth beneath an ice -sheet, 

 and of the order of magnitude of the depression of the surface 

 which it may be expected to produce. 



Let Y be the temperature at the depth .v when the surface 

 of the ground has been kept at zero temperature during t years. 



Let k be the conductivity of rock expressed in terms of its 

 own capacity for heat, the units used being one foot and one 

 year. Then, according to Lord Kelvin, /e = 400. 



If Y=f(x, t) be the equation sought, then it must fulfil the 

 following conditions : — 



(1) f(x, i) must become mx + b when £ = 0. 



(2) It must become zero when x = 0. 



(3) Since in any finite time the cooling will not bave 

 reached to an infinite depth,/(#, t) must become mx -f b, 

 when t is finite and x infinite. 



(4) It is also obvious that after the lapse of an infinite 

 time the temperature for all finite depths must become 

 mx. 



(5) The condition for conduction of heat must be satisfied, 



. dV _ d 2 Y 

 dt ~~ dx 2 



It will be found that 



X 



satisfies all the conditions ; for writing fi for — .*=., 



V4/ct 



