284 Dr. J. W. Nicholson on the Bending of 



to the sphere, or to the region of transition previously 

 defined. Since <j> n is o£ low order in comparison with <j) nr , 

 the second zero point must effectively satisfy 



—~d4>nrl'dn—0=O> 



or as (-±0) is here applicable, 



. , ax 7T n A 



so that 



m = ZiC=hax=^kr cos 0, 

 and 



- r ,(z!2±-iy . . . . cm) 



m 



and this only tends to be of lower order than z in the region 

 of space in which r cos 6 is nearly equal lo a. This has 

 already been shown to be the " transitional " region, of which 

 an investigation has been given. Thus in the "region of 

 brightness " proper, (1) is excluded, and the second zero 

 point is therefore such that m — z is positive and of higher 

 order than zi. 



Now at a zero point so defined, the non-oscillatory portion 

 R** (1 + e^Xn) of the multiplier of the exponential in the 

 series (110), which does not involve Jcr, but only z, is small 

 in comparison with any power of z' 1 . This follows quickly 

 from some results in the section " On the harmonic terms of 

 infinite order."" Accordingly, the portion of the sum de- 

 pendent on this zero point may be completely ignored*. 



This justifies the use of infinity as the upper limit of the 

 integral (119), for it is not necessary even to regard the 

 limit as of an order so low as z. Further justification is of 

 course supplied on physical grounds by the fact that, in 

 ignoring the second zero point in the earlier work, the effect 

 of a plane reflector was obtained, as it should be, for the 

 first approximation in the region of brightness. Finally, 

 therefore, we may write, with validity of subsequent appli- 

 cation, 



i 



& = 2z 



We have seen that V is finite when a) = 0. 



* In the investigation of Phil. Mag. April 1910, this zero poiut was 



j a" 



nored, bat the above analysis is desirable for completeness. 



