340 



whence 



or 



Rev. 0. Fisher on Theories to 



d 2 Y 2b 1 



dx l ~ Vn \/±ici 



z± 2x 



p 4/c£ « 



6 4**' 



b 1 z* 



_ __ t; 4nt 



X 



s/tT K 



VT K t* ' 



d 2 Y _ 1 dV 

 dx\ ~ k dt 





dY _ d 2 Y 



dt ~~ dx 2 





Hence (5) is satisfied. 



The temperature at the depth x, when the ice-sheet has 

 endured for t years, being thus shown to be expressible by 

 the formula 



X 



^— m x+— p= \ e*dfi, 

 v 7rJ e 



where zero is the freezing temperature ; since the tempera- 

 ture at the depth x before the ice-sheet was formed was 



v — mx + by 



it follows that the cooling has been 



j 2b (Viw _ . 



V TTJO 



The values of the definite integral have been tabulated. 

 They may be found in Oppolzer's Bahnbesstimmung derKometen 

 und Planeten, Zweiter Band, 1880, table x. 



Taking b as 20° and k as 400, as already explained, the 

 cooling has been calculated from the above expression for 

 various durations of an ice-sheet, and the results are given 

 in the subjoined table. It has been thought sufficient to stop 

 at depths where the cooling becomes less than one degree 

 Fahr. 



Jt appears from the above expression that the amount of 

 cooling is independent of m, the temperature gradient, and 

 that it is proportional to b, the mean temperature of the surface 

 above the freezing-point before the ice was formed. Hence 

 all the numerical results which follow, and which have been 

 calculated for a temperature of 20° above freezing, can readily 

 be adapted to any other by a simple proportion. 



