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LII. Problems relating to Underground Temperature. A 

 Fragment. By Sir W. Thomson*. 



~pROBLEM I. — A fire is lighted on a small portion of an 

 uninterrupted plane boundary of a mass of rock of the 

 precise quality of that of Calton Hill, and after burning for a 

 certain time is removed, the whole plane area of rock being 

 then freely exposed to the atmosphere. It is required to deter- 

 mine the consequent conduction of heat through the interior. 



Problem II. — It is required to trace the effect of an unusu- 

 ally hot day on the internal temperature of such a mass of 

 rock. 



Problem III. — It is required to trace the secular effect con- 

 sequent on a sudden alteration of mean temperature. 



Problem IV. — It is required to determine the change of 

 temperature within a ball of the rock consequent upon sud- 

 denly removing it from a fluid of one constant temperature 

 and plunging it into a fluid maintained at another constant 

 temperature. 



Problems I., II., and III. In solving each of these prob- 

 lems, we shall suppose the air in contact with the rock to be 

 not sensibly influenced in its temperature by the conduction 

 of heat inwards or outwards through the solid substance. In 

 reality, the stratum of air in immediate contact with the rock 

 must always have precisely the same temperature as the rock 

 itself at its bounding surface ; and the continual mixing up of 

 the different strata, whether by wind or by local convective 

 currents due to differences of temperature, tends to bring the 

 whole superincumbent mass of air to one temperature. Our 

 supposition therefore amounts to assuming that the rate of 

 variation of temperature from point to point in the rock near its 

 surface, owing to the special cause under consideration, is much 

 less than the ordinary changing variations from day to night. 

 Hence, in Problems I., II., and III., the solutions will not be 

 applicable until so much time has been allowed to elapse as 

 will leave only a residual variation, small in comparison with 



* Communicated by the Author. An old MS., written eighteen years 

 ago and found today. It was kept back until the time should be found 

 to write out the solutions of Problems II., III., and IV. The time was 

 never found ; but as mere synthesis from the solution of Problem I. suf- 

 fices for II. and III. (surface integration of the solution for I. over the 

 medial plane solves II., and the time-integral from t= — oo to t=0, of the 

 solution of II., gives that of III.), and as IV. is merely an example of 

 Fourier's now well-known solution for the globe (see Professors Ayrton 

 and Perry's paper, " On the Heat- conductivity of Stone," Philosophical 

 Magazine for April 1878), with numerical results calculated for trap-rock 

 according to its thermal conductivity, as determined by the Edinburgh 

 observations referred to in the fragment now published, the non-completion 

 of the original proposal need not be much regretted. — W. T., March 25, 

 1878. 



