212 Proceedings of the Royal Irish Academy. 



^^ p = J - 1 {cos ^ . * + sin ^ (cos <^ .j + sin ^ . /;) } ; 



^ = J- 1 {- sin ^ . «■ + cos ^ (cos <^ .j + sin ^ . h)}, 



o 



and r^ = J - 1 sin 61 {- sin <^ .y + cos ^ .k\. 



Eor instance, let the function be cos 6. Then 

 d cos B 1 



V cos 6 



dd dp 

 r-^ 



sin 



^ J - 1 {- sin . i -^ cos ^ (cos ^ ./ + sin ^ . ^) } 



The expression for the square of nabla is deduced hy the direct process 

 of multiplying together two nablas, preserving everywhere the order 

 of the operations. Thus 



^ _ / 9 1 a _i_ a _2_ ) r ^ 1 ^ Jl. 9 1 \ 



dr p W dp ddi dp } I dr p d$ dp dd) dp ^ 

 dO dcjy) [ dd dct>) 



dr^ p^ dO dd ^dA ^9p dcf) 8c^ ^sp ^ap 



3r [ dd *m dp \ p dr \ dcf) Sp In 



dd \dr p dp dd i dcj> dp ] a, 



r — V r — I r - 



dd \ a</)/ ai 



a /a_n ^]_L,1 P1A-L.\ -i. 



dcf> \ dr p) dp dcj) [ dd dp dp 



^d^ \ ^'ddj ^d^ 



