K 1 = 

 Jo 



Theory of Radiation. 645 



As we are to consider the flow o£ energy through the aperture 

 in the yz plane, we put <u = in this expression, thus 

 changing T into 



r = 27rv{t— (y sin 0sin cf)-\-z cos 0)/r}. 



But now a difficulty arises in determining the #'s, for no one 

 of the forces by itself can determine whether the wave is 

 going forwards or backwards. This difficulty is reflected in 

 the mathematics, for in the </> integration t assumes all its 

 values twice over, which prevents the usual inversion. To 

 obviate this, we separate out and add together the gh corre- 

 sponding to cf) and 7r — (f) and take the $ integration between 

 + tt/2. Thus, if 



G s =f(v,6, <j>)+g s {v, 6,ir-cf), 



we take as the typical formula 



I dv f "sin 6cW\^ </<£(Gi s cos t + G/ sin t ) . (4'3) 



Jo Jo J-tt/2 



These considerations show what formulae are aimed at. 

 To obtain them put 



„2 r P 2 r q /2 fT2 



Gr l s = 2; 2 s'mdcos(f) I dfj\ dz\ <ftS(0, y, z, t) cos r y . (4*4) 



C J-P'2 ' J-q/2 J-T/2 



and a similar expression in G 2 S an d sin t . Then, by three 

 successive inversions for (j>, 6, and v in turn, it may be shown 

 that S! = S between the limits ±p/2, ±qj2, ±1/2 and 

 vanishes outside them. Knowing the Gi's we are thus led 

 to six equations of the type of (4*2), of which the first is, 

 for instance, 



Q* = [UiO/O-UiCTr-tfO] cos^cosc/) 



-[V^+V^TT-^sin^. . (4-5) 



and there are six similar equations in G 2 , U 2 , V 2 . The IPs 

 and \ p s can be determined from these, and give 



U((/>) = {-G z cosc£ + G£sin6> + 

 V(<£) = { G y cos<£-f G Y sin6> + 



There are similar equations for JJ(tt — 6) and V(tt — <j>) and 

 two redundant equations of the form 



G y cos 6 sin c/>}/2 sin cos $\ 

 G z cos 6 sin <£}/2 sin cos <£ J 



G x =G^cos6>-G y sin<9sin(/>' 

 G a =G T cos0-G z sin 



dntfsinjl (4 . 7) 



an C7 sin <p J 



which must be satisfied identically. That they are, may be 



