Electric Waves round a Large Sphere, 159 



for the component series which formerly had a zero point. 

 On moving further across the transitional region towards 

 the shadow, this minimum becomes larger, and the region of 

 shadow may be designated as that for which this minimum 

 is no longer a small quantity. This will appear subsequently. 

 Meanwhile it may be shown that when e is small, diffraction 

 bauds must be present. 



The harmonic term whose order n is such that its exponent 

 has the minimum derivative e will obviously be the " centre" 

 of a cluster of important harmonics. The name " minimum 

 point" already suggested in an earlier section will be given 

 to this, by analogy with the theory of the zero point. 

 Denoting it by a suffix zero, it is evident that 



v f = e, i? " = 0, v = v + e(x — x ) -f ^vj"(x — -%) 3 

 in its vicinity, and the magnetic force will be derived from 



y = Xiu(l + e 2 ^)e l2V , 

 where u has the value in (66), and 



ZV = <$)n — <f)nr ~ HlO. 



In the calculation of v and u Q the expansions of the Bessel 

 functions suitable for the case in which argument and order 

 are nearly equal must be employed. These have been given 

 by the writer* as follows: — 



if r°° 



/i = 1 dw cos (la 3 + piv) l 



Jo 



/ 2 = I dw sin (w*+pw) » , . • . (70) 



Jo 



/, = f awe-*** 



Jo 

 so that/! is an Airy's integral f. 



Then if p = (m-z) {6/z)$, and m—z is not of order z, 



J m (z) = 7T-\6/z)if 1 (p) I 



J. m {z) = ^-\6/z)i {f\ (p) cos rmr + {f 2 + f 3 ) sin wrr} J ' * ' 

 and in the present case, since m = n-f-i where n is an integer, 



J—W=f-) , ^ 1 (6A)*(/i+>». • - (72) 

 the functions / having an argument p. 



* Phil. Mag. Aug. 1908. 



f Airy, <! a nib. Phil- Trans, vi. p. 379; viii. p. 595. Stckes, Math, 

 and Phys. Papers, ii. p. 329 et seq. 



