CHAP. VII.] HEAT LOST AND TRANSMITTED. 321 



. sin ml cos nl e~ x ^ mi+n *, 



and the integral h I dx Idz v gives 

 a 



cos ml sin 



. 

 n v m 2 + n 



Hence the quantity of heat which the prism loses at its surface, 

 throughout the part situated to the right of the section whose 

 abscissa is x, is composed of terms all analogous to 



sin ml cos nl + - cos ml sin 



in nl\ . 

 } 



On the other hand the quantity of heat which during the same 

 time penetrates the section whose abscissa is x is composed of 

 terms analoous to 



sin mlsiD.nl ; 

 mn 



the following equation must therefore necessarily hold 



sin ml sin nl = . sin ml cos nl 



H cos ml sin nl, 



or k (m z + ?i 2 ) sin ml sin nl = hm cos mZ sin nl + hn sin ml cos wZ ; 

 now we have separately, 



km? sin ml cos wZ = ^/?i cos ml sin wZ, 



m sin ml h 



or i- = 7 5 



cos mZ k 



we have also 



A;?i 2 sin nl sin mZ = hn cos nZ sin mZ, 



n sin ??Z A 



or r = 7 . 



cos ?iZ k 



Hence the equation is satisfied. This compensation which is in 

 cessantly established between the heat dissipated and the heat 

 transmitted, is a manifest consequence of the hypothesis ; and 

 analysis reproduces here the condition which has already been ex- 



F. H. 21 



