3 / 



-M 



Absorbing Powers of different Bodies for Light and Heat. 9 



values which those magnitudes have according to the equations 

 above obtained, while x x and y l are constant, and # 2 ?/ 2 have all 

 the different values which answer to the different points of the 

 projection of surface 2 on the plane # 2 y 2 ; the integrals accord- 

 ing to x x y x must then be taken over the projection of surface 1. 

 The double integral, 



§§dx 3 dy 3 , 



so limited is 



mCttgk 



or applying the equations for x 3 and y 3 , 



3 2 T 3*T S*T a*T \ 



$x x $x* ' 9jf , fy 2 &r, %, ' fy 2 Ss, / 2 ^' 



where the integrals are to be taken over the projection of sur- 

 face 2. Whence 



where the integrals are to be taken over the projections of sur- 

 faces 1 and 2. 



If the magnitude K' be treated in the same way, it being remem- 

 bered that a ray takes the same time to pass between two points 

 down the same path in either direction, the same expression will 

 be found for K' as for K. The proposition to be proved is thus 

 demonstrated, subject to the limitation already mentioned. This 

 limitation may, however, be got rid of by means of an observa- 

 tion made by Helmholtz in his f Physical Optics/ p. 169. Helm- 

 holtz here says (with somewhat different notation), " A ray of 

 light proceeding from point 1 arrives at point 2 after suffering 

 any number of refractions, reflexions, &c. At point 1 let any 

 two perpendicular planes a lt 6, be taken in the direction of the 

 ray ; and let the vibrations of the ray be divided into two parts, 

 one in each of these planes. Take similar planes a 2 , \ in the 

 ray at point 2 ; then the following proposition may be demon- 

 strated. If when the quantity i of light polarized in the plane 

 a x proceeds from 1 in the direction of the given ray, the part k 

 thereof of light polarized in « 2 arrives at 2, then, conversely, 

 if the quantity i of light polarized in « 2 proceeds from 2, the 

 same quantity k of light polarized in a 2 will arrive at 1*." 



* This proposition of Helmholtz ceases to hold good, as he himself 

 observes, when the plane of polarization of the ray suffers any alteration 

 such as that produced by magnetism, according to Faraday's discovery. In 

 what follows, therefore, the effect of magnetic force must be excluded. 

 Helmholtz limits his proposition also by the supposition that light suffers 



