of the Mathematical Law of the Radiation of Heat. 107 



mal to the surface, the resulting effect would be double what 

 it really is; for it would then be represented by ag instead of 

 \ag or ah. 



6. From these principles may be deduced some curious and 

 important consequences : but we proceed at present to consi- 

 der a particular case, which puts in a strong point of view the 

 necessity of the law of the sines. 



7. Let us inquire what would be the final temperature ac- 

 quired by a spherical molecule placed in the centre of a sphe- 

 rical surface having a radius R, which we conceive to be con- 

 stantly kept at the temperature a ; and continuing to call h 

 the radiating power of the surface both of the spherical in- 

 closure and of the molecule, of which we may call the radius 

 r, we shall have, as we have just seen, 



h = gfdV cos <p.f(<p) (a) 



denoting by <f> the inclination of the rays as before. 



Let oo be an infinitely small portion of the interior spherical 

 surface. It will constantly emit rays of heat which may be 

 conceived in unity of time to fill a hemisphere having a ra- 

 dius R : now the rays normal to the interior spherical surface 

 will necessarily fall upon the central molecule, and will occupy 

 upon the surface 2 it . R- of the hemisphere a space equal to 

 * r~. Hence these normal rays, all which have the intensity 

 g, and of which co is the base, will transmit to the central mo- 

 lecule a quantity of heat expressed by 



* r * /,\ 



a - a s-^w--- <« 



If in this expression we put for g its value found by equa- 

 tion («), it becomes 



a r 2 h 

 " • "2RT • fdf cos tJto) • • ■ • (2) 



and as the ratio of the whole spherical surface to w is ex- 



4> 7T R* 



pressed by , if we multiply this ratio by the expression 



(2), we shall have the whole quantity of heat received by the 



molecule, denoted by 2t. a r 2 . >+h — ^ — r, the limits of 



J fd<p cos <p.f{<p) 9 



the integral being always and \ n. 



Let us now suppose that the final temperature acquired by 



the molecule is represented by b ; it follows that the molecule 



will dissipate from its surface a quantity of heat equal to 



P2 



