544 HISTORY OF THERMOTICS. 



This proposition may at first appear strange 

 and unlikely ; but it may be shown to be a neces- 

 sary consequence of the assumed principle, by very 

 simple reasoning, which I shall give in a general 

 form in a note 24 . 



This reasoning is capable of being presented in 

 a manner quite satisfactory, by the use of mathe- 

 matical symbols, and proves that Leslie's law of 

 the sines is rigorously and mathematically true on 

 Fourier's hypothesis. And thus Fourier's theory 



4 The following reasoning may show the connexion of the 

 law of the sines in radiant heat with the general principle of 

 ultimate identity of neighbouring temperatures. The equili- 

 brium and identity of temperature between an including shell and 

 an included body, cannot obtain upon the whole, except it obtain 

 between each pair of parts of the two surfaces of the body and 

 of the shell ; that is, any part of the one surface, in its exchanges 

 with any part of the other surface, must give and receive the 

 same quantity of heat. Now the quantity exchanged, so far as 

 it depends on the receiving surface, will, by geometry, be pro- 

 portional to the sine of the obliquity of that surface : and as, in 

 the exchanges, each may be considered as receiving, the quan- 

 tity transferred must be proportional to the sines of the two 

 obliquities ; that is, to that of the giving as well as of the receiving 

 surface. 



Nor is this conclusion disturbed by the consideration, that all 

 the rays of heat which fall upon a surface are not absorbed, some 

 being reflected according to the nature of the surface. For, by 

 the other above-mentioned laws of phenomena, we know that, 

 in the same measure in which the surface loses the power of 

 admitting, it loses the power of emitting, heat ; and the super- 

 ficial parts gain, by absorbing their own radiation, as much as 

 they lose by not absorbing the incident heat ; so that the result 

 of the preceding reasoning remains unaltered. 



