546 
PROFESSOR H. LAMB ON" ELECTRICAL 
The last line of (28) is not a convenient expression when n is negative, 
readily deduce from (30) 
But W8 
(119); 
and by successive applications of this formula of reduction we find 
(-) ! T 2a+1 / d \”cos£ 
i.3... 2 n-l\tdy % ' ■ 
( 120 ); 
since, by (28), V'_ 1 (£) = cos £• This result, like (28), has been given in somewhat 
different forms by various writers.* When r is large it gives 
= iITX^i cos { jr+ n f). (121) - 
In free motion X is real and negative. We may therefore write j— — i\/v=iy where 
y is real, and may be taken positive. Substituting in (118) and (121), and expressing 
that the terms in e yr must disappear from (117), we are led to the following relation 
between the surface values of X„ and X_,,_ 1 
3.5 . 2»+l.X.-t u U ; 2a _ 1 X_„_ I =0.(122). 
Similarly in the free motions of the second type we must have 
( ?Pt ') 2re+1 
3.5 .. . 2n+l.a K -i— u 2n-i n ~' l - 1 = 0 .( 123 )- 
The equation to determine the various values of X is to be obtained, in the first 
type by elimination of X„, X_„_ 1 between (107), 108), and (122), and in the 
second type by elimination of co n , fl n , between (115), (116), and (123). 
In all practical cases j’R is exceedingly small. If we neglect all powers of yR 
above the second we have 
X*=o, n„=o. 
In the first type we then obtain 
. (124)> 
approximately. For a first approximation £R=$, where & is a root of t// / ,_ 1 (^) = 0; 
and for a second 
. (125) - 
* See C. Niven, Phil. Trans., 1880, p. 126. Also Heine, ‘ Kugelt'unctionen,’ t. i., § 60. 
