56 Proceedings of the Royal IrhJt Academy. 



generally the m system are given by maJQinia) = kj^{ma). For 

 let ^2 be two roots of the m system, then 



(1) j" rJi[mir)Ji(m2r) dr = 0. 



For 



C" 1 m' 



rJi{mir)Ji{m2r) dr = aJ{m^a)Ji{m2a) + — rJQ{m^r)JJjnir) dr 



Jo ^^'i ^ Jo 



= - — aJJ^m^ay y^m^a) + | rJJ^7nir)jQ{'m2r) dr ; 

 .'. {m^ - nii^)^ rJi{mr)Ji{m'r) dr = m<2,aJQ{moa)Jy{myd) 



where the right-hand side vanishes in virtue of the condition 

 satisfied by the m. 



(2) rJ,%mr) dr ="1 J,\ma) + f J - ^- j 



rJiimr) dr = — — Ji(ma)JJima) + rJKmr^dr 

 Jo ^^^^ Jo 



= JJ^md)Ji{ina) + — J^{ma) + m r'JJ^mr)Ji{mr)dr 



^ Jo 



which assumes form above in virtue of the relation 



ma JJ^ma) - hJy[ma) = 0. 

 In the present case, it will be seen that = 2, so that 



rJx\mr) dr = - Ji\ma). 



Let, now, any function <fi{r) be supposed to be expressed in the 

 form 



<^(r) = c^r + '^c,„Ji{mr), 

 where it will be noted that the first term corresponds to m = 0, the 



For 



