Mr. HOPKINS, ON THE MOTION OF GLACIERS. 73 



— = ,7 nearly 

 b 



Dr''' 



&c. = &c. 

 A year is taken as the unit of time. 



26. In the preceding investigation the sphere has been supposed to have a complete sliell 

 of ice. The result will also be sensibly the same if, instead of the whole surface of the sphere 

 being covered with ice, a small portion only of it be so covered, provided the thickness of the 

 ice be small compared with its superficial dimensions. This is the actual case of a glacier, to 

 which therefore equation (.S) will be approximately applicable. Let us proceed then to the inter- 

 pretation of that equation. 



We observe that when x — a few multiples of a, the value of the periodical terms becomes 

 insensible, on account of the exponential involved in them. Let x, be the least value of .t for which 

 we may neglect these terms. Then, if u^ be the temperature at that depth, 



V C 



+ V„ + yx, 



o 



ir 



= - + yj;^ (4.), 



neglecting the small quantity i\. Consequently the temperature at a certain depth is independent 



C V 



of annual variations, and lower by than it would be if the exterior shell were composed of 



• ■TT 2 



rock instead of ice ; for, in that case the value of B (Art. 2.3) would be the mean external tem- 



, , y C 



perature V, nistead of . 



' 2 TT 



If J'a be the depth for which the temperature = 0, we shall have 



V C 



= + 7^2, 



Z TT 



■■■-i(f-j). « 



which, if we give to y the value above stated (Art. 25), will be the numerical value of x.^ in 

 metres. 



If X.;, be less than the thickness of the glacier, the formula (3), and therefore (5), will be no 

 longer applicable; for (3) would give the temperature of the ice at depths greater than x,^, higher 

 than zero, which from the nature of ice is impossible. In such cases the lower surface of the 

 ice, at whatever depth it might be, would be necessarily at zero, because the heat which, if the 

 superficial crust were not ice, would elevate its temperature, will be employed in melting the ice 

 at its lower surface, which will thus be kept at the zero temperature. 



With the value of -y above given, equation (.5) gives the value of x, supposing the ratio of 

 the conductive power of ice to its specific heat to be the same as for the rocky crust of the earth. 

 If this be not the case, the eciuation (5) will still give the depth at which the temperature = zero, 

 by assigning the proper value to y as depending on the ratio just mentioned for ice. 



As a numerical example, suppose V = 0, and C = 15" (cent.) We shall have at the dejith x 



w, = - 5" nearly ; 



and X., = — — feet, 



,028 



= 178 feet nearly. 



Vol.. VIII. Paht I. K 



