Mr. G. W. Hearn on a permanent State of Heat. 23 



Now the extreme temperatures being represented by 1, it 

 is evident that t^ will be some proper fraction. Let there- 

 fore 



/ 2 = sin «, 



ka a / '" ci 



.♦. £2 = cot—, and e~ * =tan— •. 

 2 2 



Hence 



^Y^cot-^tan-^ + ^tan^ - |, 



where n = — . 



a 



Now let t 3 be another observed temperature at the distance 

 of — - a from either extremity, 



whence tan 



2 - / 3 3 



so that from those two observed temperatures we can easily 

 calculate a, and thence v. 



To adapt the formulae to logarithmic calculation, let 



f - sin * 



*3 



then tan — = 8 sin —. 



2 2 



Moreover, make 



(tan— -I = tanfl, .\ ( cot— I =cot0, 



then v = t<2 cosec 2 0. 



Hence we have only to calculate 

 log sin i] a log / 3 — log ^ 3 , 



log tan ss (1 — 2 ») -J log 8 + 6 log sin -| L 



log v = log / 2 -f- log cosec 2 1 



ha g. 



It is also evident, since e^ — cot — , we can obtain Jc, and 

 thence v^ the ratio of the exterior to the interior conductivity. 



