Electric Waves round a Large Sphere. L63 



the type which oscillates. Now e is positive, and therefore 

 for oscillation to take place it is sufficient that v "' should be 

 negative. This condition is evidently satisfied, for v Q " has 

 decreased from its positive value in the region of brightness 

 to a zero value, so that its derivate is continuously negative. 

 Otherwise, the result follows from (80). The existence of 

 maxima and minima is therefore demonstrated. But on 

 passing further into the region, towards the geometrical 

 shadow, e tends to increase, and the terms near the minimum 

 point no longer have a preponderant sum. Moreover, 

 % n tends to behave like <£». The oscillation between maximum 

 and minimum ceases, and in fact the whole effect becomes 

 of a smaller order of magnitude, the series hitherto neglected 

 contributing to an equal extent. When e is of higher order 

 than 2T~f, the term of the exponential involving Vq" 

 becomes unimportant, and the sum is at once of lower order 

 in so far as it depends on the series hitherto most important. 

 It will appear later that the series tend to cancel one another 

 in a remarkable way. 



The above investigation is restricted to points in the 

 transitional region not too far away from the obstacle, for 

 <f> nr has been assumed to be of the same type as (p n for a 

 given value of n. Thus the bands will disappear at a sufficient 

 distance. The investigation for the small values of 6 is 

 postponed, and the succeeding sections take up the problem 

 of the geometrical shadow. 



Preliminary discussion of the geometrical shadow. 



In the geometrical shadow, as we have seen, it is not 

 possible for the derivate of an exponent, in any of the series 

 for the magnetic force, to vanish or become small. No 

 group of terms, therefore, becomes of supreme importance 

 after the manner of previous sections. Moreover, as was 

 pointed out earlier, it follows that the harmonics of low order 

 may contribute substantially to the sum, and thus the 

 asymptotic expansion of the zonal harmonic must not be 

 used even for a finite orientation from the transmitter. The 

 proper formula was given in (48) and becomes 



yp = G(P) 5 m{K n n m yHl + e 2l ^ysinmcj,e li ^-^^ 



71 — 1 



where G (6) is an operation defined by 



Gr(0).w= — ri — a \ - , j . (8.-5) 



v y ka-nr dvj e v/ 2V (cost/ — cos<£) 



M2 



