of the Mathematical Law of the Radiation of Heat. 105 



The quantity of beat which reaches y from a will be identical 

 with that which reaches A from /3, provided A/3 = «y. Now, 

 by the hypothesis, the abscissa j3 n represents this quantity. 

 Next let us take (fig. 2.) A p B y , C, and a. i as before, and erect 

 the abscissa <x t n t equal and parallel to (5 n, and let us repeat a 

 similar operation for all the other points of the line ay; we 

 shall then have another curve C, n, o i . . . of which the surface 

 will represent the intensity of a ray inclined to the normal. 



In order to compare the surfaces of the curves fig. 1 and 2, 

 we have only to observe, that for the same abscissa A x = /3 n 

 (fig. 1.), or A,x, = a,?*, (fig. 2.), if the ordinate (fig. I.) is re- 

 presented by A/3 = ay, and we make ay = 1, the ordinate 

 A / a t (fig. 2.) equal to A a will be represented by sin p. But 

 when two curves have the same origin and axis of abscissae, 

 their surfaces comprised between the origin and a common 

 limit are evidently in the relation of the respective ordinates : 

 therefore, the surface AC no: surface A / C, n / o / = 1 : sin <p. 

 Wherefore the respective intensities of a ray of heat \l normal 

 to the surface, and one, v inclined at an angle <p have the 

 same ratio, and v = \l sin <p. q. e. d. 



2. This demonstration is equally applicable to curve as to 

 plane surfaces. For the thickness of the physical surface CA 

 being extremely small, the portion of the mathematical surface 

 included between the extremity of the normal and that of the 

 oblique line (which at most can only be equal to the length of 

 CA), will always be sufficiently small to be confounded with a 

 plane tangential to the point of emission. 



3. The absolute intensities which are supposed to be known 

 in the preceding demonstration are in no respect wanted for 

 the determination of the intensity of the oblique rays rela- 

 tively to the normal ray, 



4. Having demonstrated the law of radiation upon these 

 simple principles, we proceed to show, that did this law not 

 exist we should arrive at conclusions at variance with the 

 simplest experiments. But we must first introduce a distinct 

 conception of the radiating power of a given surface. 



5. Let a be the temperature of a heated surface, and h its 

 radiating power*. Each infinitely small portion of the sur- 

 face may be viewed as the centre of a hemisphere which is 

 filled by the radiant heat emanating from it. If then we con- 

 sider a small portion of the surface taken as unity, the quan- 



* This coefficient h depends on the nature of the radiating body, and is 

 what Fourier calls " Conducibilite exterieure " in his Trait'e Analytiquc dc 

 Ckalcur. — Tuanslator. 



Third Series. Vol. 2. No. 8. Feb. 1833. P 



