306 Prof. W. B. Morton on the 



Then 



\ 



IT 



a== - - + arg. m-arg. (<r) i 



.... (9) 



£ = _£-arg.( C 2 ) | 



2 



Using equation (8) and remembering that arg. (& 2 «)= — , we 

 Jiave 



-»■«— I "«8-^g. • • • (10) 



B=arg -J7(M-I + ar s- m 



Jl(&2°0 T 



Jo(& 2 a) 5 

 and /3 = arg,4# a |-T 



(11) 



J (k 2 a) 



- to exa 

 for different values of the variable, and also arg. m 



>fc 



J, 



It is now necessary to examine the values of are. T , 7 2 - 

 ss of the variable, and also arg. m. 



Taking first — ^ we have for small values of k 2 a 



Jo 



J, 



— 2^'2 ff 5 



the argument is j, and /3 = 0. 



For very large values -rJ = /, giving ft=j • To trace the 



course of the magnitude ft between these limits we can use 

 the tables of J ( t yv/i) and Ji(.r^i), which have been com- 

 puted by Aldis *. His argument x corresponds to 2a* / ^E^^ 

 and runs between values 01 and 6*0 ; the corresponding 

 values of arg. ~ come out 44° 23' and 86° 10'. The curve 



marked ft in fig. 1 is plotted by calculation from Aldis's 

 tables. 



* Aldis, Proceedings of Roy. Soc. vol. lxvi. pp. 42, 43 (1899). 



