Electric Waves round a Large Sphere. 529 



Thus 'dcfrj'dn is negative. As n increases, it tends to the 

 order m— i. 



If 2— m is of lower order than zi, 



and is therefore again negative, and of order m~3 with 1/R n , 

 on both sides of the critical point m=z. 

 When n is greater, writing m = z cosh ft, 



*t n ftn=-/3 by (29). 



But tan<£„ = e 2 S 



and therefore B$n/d^ is again negative. Moreover, it 

 decreases rapidjy as n increases, on account of t n . 



Finally, 'd(f> n /'dn is always negative, as must also be 

 "dcf) nr /'dn. Since, moreover, they are of order not greater than 

 m-% except in the first region of expansion of the Bessel 

 functions, it appears that the four functions v cannot have 

 zero derivates except in this region. 



Thus in finding such derivates, we may write, if t c = n + i/s 



4> n = z< (1 — .r^ + A'sin- 1 ^— ~x > -f ~ 



cf> nr = ^(c-'-jfy + xsin-icx- |#j + £ 



where c denotes a/r, and the four derivates become, rejecting 

 Xn, which is very small, to 



Vi = z(sin -1 x — sin"" 1 ex + 0), 

 v 2 ' = z(sin- 1 t v — sin~ 1 cx—<l)), 

 v/ = ^(sin -1 ^ — sin -1 cx + (j>), 

 v/ = z(sm~ 1 .c^sin- 1 c.v — cf)). 



Since c is not greater than unity, sin -1 or cannot exceed 

 gin -1 x for a positive value of x (corresponding to an harmonic 

 of positive order), and therefore v/ and vj can never be zero, 

 vj and u 4 ' have the same vanishing point, given by 



sin 



« + #, 



or «r = ca? cos <£ + sin cf> \/l — c 2 ^ 2 , 



leading finally to 



<r = sin 0/(1 — 2c cos + c 2 )i. . . . (41) 

 Phil Mag. S. 6. Vol. 19. No. 112. J^ri/ 1910. 2 M 



