646 



Mr. C. G. Darwin on the 



seen from the fact that, expressed in terms of the forces X 

 etc. by (4'4), they are simply those two of the electro- 

 magnetic equations which do not involve the operator d/d# — 

 that is, which are valid on the yz plane. 



5. We next find the flow of energy through the aperture, 

 and apply Rayleigh's theorem to it. The flow in the positive 

 direction of x per unit area per unit time is the component 

 of the Poynting vector (Y<y — Z/3)c/47r, and so the total flow 

 through the aperture in time T is F, where 



c Cp' 2 C^ 2 C t > 2 



= a— \ dy \ dz \ dt \ dv \ sin e de \ d( t> 



^^J-pl2 J-q/2 J-T/2 Jo Jo J-ir/2 



\ dv ] \ sintf'dfl'l d(f>' 



Jo Jo J-t/2 



[{G] Y cos r + G 2 Y sin ToKGx 7 ' cos r ' -f G 2 /! sin t '} 

 -{Gi z cos r + G 2 z sin ToHGV 3 ' cos Tq' + G/' sin t '}] 



In this G^" is the same function as G^, but with argu- 

 ments v\ 0' , <j>' 9 and t ' is the same as t , but with arguments 

 v', f , </>', y, z, t. _ 



To avoid writing out many terms which will afterwards 

 vanish in the integrations, observe that in this expression 

 only those arising out of J cos (t — t</) will contribute any- 

 thing, and that those consist of the sum of identical terms in 

 GJs and G 2 's. We shall therefore only consider the former. 

 Reverse the order of the integrations and perforin those for 

 y, z, t ; then 



(5-1) 



F=|^2 y dvddd<f>dv' d0' d<f>' sin 6 sin 6' {G^G! 



G^G/'} 



sin nr{y' — v)T sin 77- (V sin 0' sin </>' — v sin sin typ/i 

 7r(v' — v) tt(V sin 0' sin <£' — ysin sin <j>)/c 



sin 7r(V cos 0' — v cos 0)g/c 

 ir(y cos 0' — -v cos 0)/c 



(5-2) 



Now integrate for z/. This removes the first factor of the 

 second line and changes v' into v in the rest. Next, the 0' 

 integration removes the factor in the third line, giving a 

 multiplier r/^sin and changing 0' into in the rest. Then 



