106 M. Maurice's Abstract of Fourier's Demonstration 



tity of heat radiated by it will be proportional to the product 

 ah; and if we could know how much heat traversed in unity 

 of time the surface of a hemisphere of radius 1, having for 

 its centre an element of surface taken for unity, we should 

 have the value of h by dividing the expression for that quan- 

 tity by the product a . 2 jr. 



In order to determine this quantity, let us designate by g 

 the constant coefficient which represents the intensity of a ray 

 of heat normal to the surface. If this intensity varies with 

 the inclination <p of the rays, we may represent it by gf(<p)> 

 where f (<p) denotes an unknown function of the inclination. 

 Hence agf(q>) will represent the heat afforded by a ray 

 making an angle <p with the surface. 



Let us next consider upon the hemisphere of radius 1, an 

 elementary zone which has for height the element d 4> of the 

 arc <p, and for base the circle 2 ■* . cos <p : it is evident that the 

 product ag ,/(<p). 2 7r cos <pd$ will express the quantity of 

 heat which in unity of time will traverse the surface of the 

 elementary zone ; and consequently the integral of this ex- 

 pression taken from <p = to <p = J x, will express the 

 quantity of heat which proceeding from unity of surface will 

 traverse in unity of time the hemispherical surface 2 *■. But 

 this quantity ought also to be exactly represented by a li 2 ?r. 



Hence 2 am . h = 2a7r.gfd<p cos <p .f{<p) 



or, more simply, h = gfd <p cos <p ,f (<p), 



taking the integral between the limits first assigned. Such 

 will be the general expression of the radiating power of a given 

 surface. 



Thus, for example, if the intensity of the rays be the same 

 for all angles of inclination, wehave t /(<p) = 1, and integrating 

 the expression for li between the given limits, we have h = g, 

 as it ought to be upon this supposition. 



If, on the contrary, as we have seen in article 1, the inten- 

 sity is proportional to the sine of the angle of emission, we 

 shall havey*(<p) = sin <J>, which gives h = \ g. Hence in the 

 case of nature, in which the general intensity of the rays is ex- 

 pressed by g. sin <p, it has for extreme values zero and g; and 

 the mean value of k, the radiating power, is ±g. Such would 

 be the intensity of rays emitted at an angle of 30°, for, 



We also see that if all the rays were similar to those nor- 



