( 739 ) 



if Uo is the value of u for tlie direction OQ. Indeed the dii-ections for 

 wliich ?< = const, lie on a cone surrounding the line OQ, just because 

 M is a maximum for that line. We first integrate with respect to ip 

 and put 



sm (p — dxp = J (it) (12) 



-I^ 



T'Vcof» 



du 



The result is 



(5j := j ƒ (m) sin (gu — h) (Jti 



(13) 



^ 10. An integral such as (13) has already been considered by 

 KiRCHHOFF. For great values of g it approaches nniformly to zero 

 and at infinity it may be represented by a development of the form 



a, a. 



-+^+ ... 

 9 9 



It is only the coefficient a, that can be found. Ijitegration by parts 



of the integral gives 







I / {u) sin (gu — h) du = 1 1- . . (14) 



J ' ' 9 9" 



"0 



The first term, taken by itself, gives a sufficiently exact result for 

 points P, lying at distances r from 0, which are large in comparison 

 with the wavelength of light ; in the following development we have 

 in view only such points as satisfy this condition. We put therefore 







/("o) COS {gu^—h) —f{o) COS h 



ƒ 



f{u) sin (gu — A) du ^ ' 



9 



(14«) 



We shall first consider the part 



Now 



f (0) cosh TV, ^ t rr 7iab,T , d<p 



= cos l:i-- I ——- sm $ — - 



Ig 2nr TJ |_T» V' cos 9- * du 







d(cos$) d{cos^) 



u = 



bin (p —- ■= — . r — , 



du o(cos5) ou 



du 



ö(eo«S) ö(co«S) 

 so tliat for ?/ = or cos ? = 





