Electric Waves round a Large Sphere. 759 



A first order value only of the second integral, 



j _C 9 d<f> sm </> 3 -f 8 sin 2 eft ,% a cos ^ H71) 



2 Jo Vcos^-cosO* cos 3 ^ 



is required, The important part of this integral is con- 

 tributed at the upper limit, and is given by 



f = 3 + 8 sin 2 C e d^'mef) ^« O5S 

 cos 3 # J \/eos(/> — cos# 



or 



Hi) 



7ry3_+8_sin 2 ^ .JcaQo&B+h* 

 cos 3 



and to the same order, 



so that to a second approximation, 'with a relative order 



_ 2& sin _ t ^ r _i t v /^Ii _i_ dl_ 2 \ 



7P ~W7^a e ^ d6 + ™*.d&} 



«* -2^ sin 2 fl(l f ~) ,-^+^cos^ 



where 



g = (3 + 8sin 2 0)/12cos 3 0. . . . (173) 



This is identical with (155), and has been shown to be 

 valid so long as z sin \Q or \z6 is small. If z is 10 6 as in 

 our usual numerical case, £#=10 if 6 is about two seconds. 

 In other words, the result is valid actually on the axis, and 

 the solution does not change in type near that axis, in 

 accordance with the statement at the end of the preceding 

 section. The actual effect is practically evanescent at such 

 a small orientation. 



General investigation of the Shadow, 



The only remaining case which demands solution is that 

 of tbe intensity at any point in the shadow, and to this we 

 proceed. The intensity for any point on the surface of the 

 sphere has been tabulated already, and the results for any 

 other point in the shadow may be derived very readily, after 

 the same manner, from preceding analysis. The problem 

 thus contemplated includes, of course, that of a raised 

 receiver of the waves. 



