8 Mr. Hopkins on the Motion of Glaciers. 



to be represented by V + C cos f 2 tt / + -|- ) . So long as this 



is less than zero, the problem will present no peculiarity ari- 

 sing from the circumstance of the exterior crust being com- 

 posed of ice; but however greatly the external temperature 

 may exceed zero, the superficial temperature of the crust can- 

 not, from the nature of ice, rise higher than zero. Hence, 

 while the external temperature is below zero, we shall have 

 the ordinary case of a solid body placed in a medium of which 

 the temperature varies according to a given law ; but when the 

 external temperature rises above zero, the condition at the sur- 

 face will be that the superjicial temperature of the mass shall 

 be constantly at zero. Instead of this last condition, however, 

 we may suppose that, during the time it would hold, the ex- 

 ternal temperature shall be zero; for it is manifest that the 

 two conditions will in the case we are contemplating be very 

 approximately the same. Hence, then, the case for investiga- 

 tion will be that of a sphere of large dimensions cooling in a 



medium of which the temperature is V + C cos (l-Kt •\- — \ 



when this quantity is negative, and zero for those values of t 

 which render the expression positive. If V = the first of 



these conditions will be satisfied from if = to / = -— , from 



2 



3 1 



;=1 to ^=-— , &c.; and the second from t=—- to ^=1, from 



2i '^ 



t— — to t — % &c. If V do not = 0, the former of these 

 2 



periods will be shortened and the latter lengthened, or the 



converse, according as V is positive or negative; if, however, 



V be small compared with C, the periods will be approximately 



as above stated, and such, therefore, we shall consider them. 



They will be semi-annual, if we take one year as the unit of 



time. 



Let V denote the temperature which would exist at a point 

 within the sphere at a depth x beneath its surface, if the ex- 

 ternal temperature were always equal zero. We shall have 

 [x being small compared with the radius of the sphere) 



t; = t^o + y cT, 

 where Vq is the superficial temperature of the sphere, and y 

 the rate at which the temperature depending on the original 

 heat of the sphere, increases with the depth. Assuming the 

 time of cooling to have been very great (as in the case of the 

 earth), Vq will be extremely small. 



Again, let li denote that part of the internal temperature 



