314 CLAUSIUS ON THE CONSTITUENTS OF THE ATMOSPHERE 



The intensity passing through an element dy is therefore 



— k-rdy ds. 

 dx ^ 



Applying this to the alteration which has taken place in the 



element of the surface MN M'N', then while the rays travel the 



dv 

 distance ds, the intensity —k-r-dyds has passed over the 



boundary MM' into the element, and, on the other hand, the 

 intensity 



has passed out over the boundary NN', so that the excess which 

 the element has hereby gained is 



d^v 

 k-z—^dxdyds (II.) 



The two other boundaries, MN and M'N', must be treated in a 

 similar manner, and thus we find the gain of the element to be 



d^v 

 k-^dxdyds (Ill') 



ay 



Now as the total increase of intensity which the element has 

 gained by the passage of the rays through the distance ds must 



be expressed by -r dx dy ds, we obtain by combining (I.), (II.) 



and(lll.), 



-^ dx dy ds= —hv dx dyds-\-k -r-^ dx dy ds + k ^-^ dx dy ds, 



from which we derive the partial differential equation sought : 



S-K£-S)-^-« (^) 



By an easy substitution this may be further simplified. Setting 



v=u,e-% (4) 



it becomes 



As this equation quite coincides with the former, with the ex- 

 ception that the last term is absent, we see that u is the bright- 

 ness obtained for the point M afler the rays have passed over 



