s 



Electric Waves round a Large Sphere. 283 



region of brightness, the first approximation being sufficient 

 for a proof. With this convention, v is a minimum value 

 of w, and since v is negative, v — v must always be positive, 

 there being no other stationary points of v. 

 Let a new variable co be defined by 



»=(*— v )i (118) 



Then co must always be real. The sign of the ambiguity 

 in co is to be identical with that of as— # , which is of 

 necessity a factor of &> 2 . Let g>=— 8 correspond to x=e. 

 Then, since v ! dx = 2codco, 



2ze lzv °C dcoYe lZ( °\ . . . (119) 



where Y—TJco/v' expressed as a function of co. This function 

 will not be oscillatory. In the physical problem, of course, 

 the value of 8 may be found on substitution of the proper 

 values of <\> n and cp w in v. It is sufficient for our purpose to 

 observe that it is not small. 



V is finite when ft> = 0, corresponding to x = x Q . For U is 

 finite, and the limiting value of co\v' is that of 



{W [x-stfWim-ajvJ' or (2i*")H 



which is also finite, since v " is not zero. This limit is of 

 course also real, for v " is positive. 



But the selection of infinity as the upper limit of the 

 integration with respect to co calls for further remarks when 

 the results are applied afterwards to the actual case. For 

 when x is infinite, cf> n and cp ni . both vanish, so that v — v Q 

 becomes —0x — v Q} which is negatively infinite. It follows 

 that another point has occurred, beyond x = ,x , where v f is 

 zero, so that the present assumption of only one zero point 

 in the range will not strictly correspond to the case for 

 which its application is intended. But it was shown in an 

 earlier section that no second point of this kind, in addition 

 to a' , can occur when m is such that cf> n and <fi nr are both of 

 order z, or when both are of lower order than z, whether in 

 the range of values for which they are approximately -^7r or 

 beyond. It must therefore occur when the smaller only is 

 of lower order than z, and not the larger. In other words, 

 it occurs for a value of m such that kr — m is positive and 

 of order kr, but at the same time either (1) z — m or m—z is 

 only of order zi or less, or (2) m—z is positive and of order 

 higher than zi. 



Now let attention be restricted to points of space not close 



112 



