210 Proceedings of the Royal Irish Academy. 



Next apply them to find v^^"N 



P 



^V" = m(m - 1) r"^-"^ — + 2^/-"'"^ 1 — I -, 

 P VpV ^ 



= m(»* - 1) r'""^ — + 2mr'""^ -, 

 P p" 



- m{m + 1) r'""^ — , 

 P 

 which is the correct result. 



Eor any function of r and p, whether scalar or vector, 



9( ) 3( ) 



dr p dp r 



Hence for any such function, 



^_ a 1 a 2W 9 1 9 2 



dr p dp r ) \dr p dp r 



d fd{ ) 1\ HD 3 fd{ )2\2W 

 dr\ dr pj p dp\ dp rjr 



d_/d( )2\ 1(3) d /d{ ) n2W 

 dr\ dp rjp dp\ dr pjr 



_d\)\ t') d\ )4 (2) 

 dr"^ p^ dp'' r^ 



d\ ) 2 (3) a( ) 2 W 



+ 



3r3/3 rp dp r^p 



d^{ ) 2 (5) an 2 («) 



a^ar pr dr p^r 



If the function is a function of r only, all the terms involving- 

 vanish, and v^ becomes 



a^ni^sni. 



ar- p^ ar p^r 



a 

 dp 



