24 Mr. G. W. Hearn on a permanent State of Heat, 



The quantity k will be usually very great, and .*. a very 

 small ; and hence when n < — , we shall have very nearly 



v = 

 1 



£*^t). 



and when n > — 

 2 



»-■ $*(•*-£}/ 



Let v and r/ be two temperatures corresponding to 

 .-.v = -t,[cot-) , */ = -*, (tan -J 



... vx! = jijc6i-j. 



Hence the product of the temperatures of points distant 

 from each other by half the length of the wire (provided nei- 

 ther are very near the extremity) is nearly constant. 



There is another curious relation which is not merely ap- 

 proximative, but accurately true, and which is easily verified 

 by experiment. It is as follows : — 



Having, as above determined, 0, find m from the formula 



1-2 TO / V 



(tanfl) 1 - 2 " = tan ( — — 0j, 



and then s' = am; 



and if v and v' be the temperatures at 5 and a', 



v = t 2 cosec 2 0, 



i/ = U sec 2 ; 



or the sum of the reciprocals of the squares of the tempera- 

 tures at such points is constant. 



It is to be observed that Newton's law of cooling in vacuo 

 is the basis of the above investigation. Also, that as the ex- 

 treme temperatures of the wire are supposed equal, the two 

 extremities may be joined and subjected to a single source of 

 heat, in which case it will be a closed curve*. 



* Vide Fourier, Theorie de la Chaleur, and Kelland on Heat. 



