of the Mathematical Law of the Radiation of Heat . 109 



of the given space, which have the common temperature a 9 

 two extremely small portions of the surface, which are plane 

 and homogeneous, and which may be denoted by s and s*. 

 Let 8 be the distance between 5 and s f , which is finite, and 

 therefore incomparably greater than the dimensions of these 

 very small portions of surface : — we have to find how much 

 heat the surface s, for example, receives from s, in unity of 

 time ; neglecting, as we have hitherto done, the portion of 

 heat reflected, since we shall do the same when we come to 

 consider how much heat s receives from s'. 



Let us call p the angle which 8 makes with s 9 and <j> that which 

 it makes with sf. We may reckon the distance 8 from any points 

 of the surfaces s and 5', since from their small size no sensible 

 variation could be introduced into the length of 8, or the an- 

 gles p and . Each infinitely small portion ca of the surface s 

 will be the base of a ray of heat falling upon 5' ; and if, to 

 know how much heat this ray contains, we make through a 

 point of s 1 a section of the ray perpendicular to its direction, 

 we shall obviously have s' sin <p for the area of that section ; 

 a quantity of which we must take the ratio to the whole surface 

 2 it 8 9 of the hemisphere traversed by all the rays emanating 

 from «j, when we wish to measure the quantity of heat which 

 falls from s upon s'. 



Now, denoting by f(p) the unknown function of the incli- 

 nation p of the ray which determines its intensity, we shall 

 have the product a> .ag .f (p) for the heat of the pencil 

 emitted from co at an inclination p ; g representing, as before, 

 the intensity of a ray normal to the surface. Then multiplying 



this by , and also by ■— - — ^-, the ratio of the surfaces, we 



CO J 27T6- 



have the expression g ^ . sf(p) . J sin <p 



for the total amount of heat passing from the surface s to the 

 surface s'. 



But it is evident that, reasoning in a similar manner, the 

 quantity of heat passing from / to s in the same unity of time 



wiU be ~Hw ' ^^ ' s sin P ' 



It follows from the comparison of these expressions, that if 

 the unknown function/ (p) ovf(<p) be the sine of those an- 

 gles, the action of 5 will be the same upon s! as that of s' is 

 upon s ; and that if this function does not represent the sine, 

 these two actions cannot be equal. 



Hence it is easy to see that unless this condition be ful- 



