214 Proceedings of the Royal Irish Academy. 



When these terms are collected according to the differential 

 operator, we obtain 



^ _ a2( ) 1(1) 1 8( ) 1 (^) 1 3'( ) 1 f3) 



72 _ 



2^2 p2 ^: 



2 ;^/32 



r' 



9^y \b4> 



r 8r9^ {dp dp\ 



r 3/'9(^ \dp dp \ 



1 a2( ) / _]__ 1 \ (6) 



r"^ d6d(fi } dp dp dp dp I 



r dr \p^ p^ 



1 9( ) /' 1 92p 2 1__ ^1 9V JL_ys) 



"^ /^ 'W \ ^ /9pV 9^ 9^ ~ 9p "^ /9/3V 9^ 9p 

 5Z ^-ZP 



/9pV 



1 9( ) / 1 92p J_ J_ 1 9^p J_ \ (9) 



"^ r^ "9^ I ^ /9p Y 9^' dp ~ dp "^ /9p Y 9^9.^ 9p ( 



( \d^j 9^ 9^^ V90J 9^j 



/9pV fdp\^ 

 Since p'^ reduces to - 1, — to - 1, and ^ to - sia^^, the 



first three terms reduce to 



92 1 92 1 



9^2 ^2 9^2 ^^ gij2 ^■^z 9 



By the same principles of reduction, the space-coefficient of (4) reduce 

 to 



9p 9p 



