and Electromagnetic Theory. 317 



of Tallies of p ; if the range is limited, then a margin of values 

 of p will be imperfectly represented, or the relation between 

 p and p f ceases to have the uniform character given above 

 within a margin at either extreme. The extent of such 

 margin depends on r or w/V, and when r is small is inappre- 

 ciable ; but is not so if r is finite. The conclusions drawn as 

 to the fourth power of temperature require the range cf p to 

 be made sufficiently wide to ensure this penumbral effect being 

 of no importance, but need not be infinite unless w = V. 



In connexion with this question of the fourth power, if we 

 look back to the construction of (3) the use of 



p z (Vn — w)dn = p' z (Vn' + iv)dn' 



or p 2 (Vn — iv) dp dn =p' 2 (Vn r + w) dp' dn! 



seems to be essential, the further subdivision into 



p dp dn =p'dp'dn / and p (Vn — w) =p' (Vn + w) 



not being possible in refraction. As a matter of analysis, a 

 more general collective relation, involving transit with some 

 loss or modification, would be got by multiplying the above 

 by an arbitrary function of 



Jo(l — qn) or p'(l + qn f ). 



When the object is to allow for work done by pressure it is 

 prima facie reasonable to make the work depend directly on 

 wn the component of translation along the wave-normal, and 

 use the linear factor. In this way the theory is brought into 

 relation with the laws of Stefan and Wien; but the point will 

 be mentioned later. 



In reflexion two of the factors p in p^ are connected with 

 the angular correspondence in the original and reflected 

 streams, the other two are connected with the relative velo- 

 cities Vn — w and V— -urn, the former a velocity of transit, 

 the latter connected with change of period. When we pass 

 to the collective relation this is true as to the source of the 

 power in # 4 . In effect radiation being a question of exchange, 

 its law depends on the degree of freedom for exchange : for 

 the analogues in two dimensions and in one dimension the 

 appropriate powers are the cube and the square. 



For two dimensions (2) is correct, but (3) suggests a wrong 

 angular element: the element is d$ (not sin Odd, i. e. 

 sin 0d6d<j> with d(f> discarded), and 



dn : dri=d9 s/l-ri 2 :dO f s/l-n'^pdO : p'd& ; 

 so that for (3) should be written 

 p z J(V cos 6 - w)(l -r cos 0)d6=p' 3 $(V cos 6' + w){l + r cos 0')d&. 



