Ml'. Hopkins on the Motion of Glaciers. 9 



which depends on the external temperature. Then, if m de- 

 note the whole internal temperature at the depth x^ we have 

 M = t> + m' 

 = i^o + y ^ + m'. 

 The theorems given by Poisson in his Theorie de la Chnleur, 

 articles ID-i, 195 and 196, will enable us to obtain the expres- 

 sion for u'. For the investigation I must refer to my first me- 

 moir on glaciers. 

 We obtain 



V 



- — + ^0 + 1 



7r 



___ + t,^ + y.. 





Ml-) 



u Kj -Via- / X \ 



+ T^'^; — ^ — e <* cos ( 4 7r/ \/2 7r — 8,j 



b_ C -^vT; 



Dj'l.S.TT 



+ &c. 



I am not aware of any experiments for the determination of 

 a and b for ice. Poisson has given their values for the case of 

 the earth, deduced from observations made at Paris on the an- 

 nual variations of temperature at different depths. They are 



^>= 1-05719 I ^"metres. 

 He also gives 



y=0°-028l} (centigrade). 



D, Dj, &c. are constants, such that with the above values of 

 a and h, we have 



^ = -7 nearly, 



d;- ^^ 



&c. = &c. 

 A year is taken for the unit of time. 



In this investigation the sphere has been supposed to have 

 a complete shell of ice. The result will also be sensiblv the 

 same, if, instead of the whole surface of the sphere being co- 

 vered with ice, a small portion only of it be so covered, pro- 

 vided the thickness of the ice be small compared with its su- 

 perficial dimensions. This is the actual case of a glacier, to 

 which therefore equation (1.) will be approximately applicable. 

 Let us proceed then to the interpretation of that equation. 



We observe that when .r = a few multiples of a, the value 



