Mr. HOPKINS, ON THE MOTION OF GLACIERS. 69 



The determination of /3 would afford an obvious means of approximating to the thickness of 

 the glacier at any proposed point. For suppose /3 determined for all those different portions of 

 the glacier where a difference of inclination of the upper surface might indicate a corresponding 

 difference in that of the lower one. Let the length of the successive portions, beginning at the 



lower extremity, be o, a-i ffl„; and let ci a-i a„, /3i ^2 /3„ be the corresponding 



values of a and /3. Then if A, be the vertical thickness at the lower extremity, and h the re- 

 quired thickness at a distance =a^ + a.i+ a„ from that extremity, we shall have 



h = hi + ra, (tan a, - tan /3|) + + Cfi (tan n„ - tan/3„). 



The chief practical difficulty in the application of this formula would be in the determina- 

 of jS, /Sai &c. with sufficient accuracy. It appears not improbable, however, that the limits of 

 error in determining /3 by the formula above given for tan /3, would be such as to render the deter- 

 mination a sure approximation to the real value; and, at all events, if it were found impracticable 



to determine all the quantities /3i /Sj /3„, and therefore the complete thickness of the glacier, 



such of them as should correspond to the more accessible and least irregular parts of the glacier, 

 might probably be determined with considerable accuracy, and thus the rate of increase of thick- 

 ness in these parts would be known. 



SECTION V. 

 Internal Temperature of a Glacier. 



22. In a previous section I have given the general reasoning by which we conclude that 

 the temperature at the lower surface of a glacier of considerable thickness cannot be higher than 

 zero of the centigrade thermometer. Since this conclusion, however, is of the first importance in 

 the theory which has now been offered of glacial motion, I shall give the mathematical investigation 

 of the problem. The case taken for direct investigation will be that of a large sphere, like the 

 earth, of which the temperature increases as we descend, coated with an external shell of ice, the 

 temperature of the shell being at every point below zero (cent.), that the ice may in every part 

 remain perfectly solid. We shall thus be able to deduce the limiting thickness of the icy crust 

 compatible with this condition of perfect solidity. If the thickness exceed this limit, then must 

 its lower surface be in a state of constant disintegration, as already explained (Art. i). 



We have no exact knowledge of the conductive power of ice, but there is no reason to doubt 

 its being very small. I shall suppose it (for the greater simplicity of investigation) to be the same 

 as that of the earthy matter supposed to constitute the nucleus of the sphere; and for the 

 same reason I shall also suppose the conductive power from the nucleus to the icy envelope to 

 be the same as in the interior of the nucleus, or in that of the icy crust. I shall also assume the 



external temperature to be represented by V + C co& {%-wt. + — \. So long as this is less than 



zero, the problem will present no peculiarity arising from the circumstance of the exterior crust 

 being composed of ice ; but however much the external temperature may exceed zero, the superficial 

 temperature of the crust cannot, 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 surface will be that the superficial 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 external temperature shall be zero; for it is manifest tliat 

 the two condtions will in the case we are contemplating be very approximately the same. Hence, 

 then, the case for investigation will be that of a sphere of large dimensions cooling in a medium 



