RoUKiiTS — The SijinbolicitL Bxpressioii of Eliiiiiuunis. 73 



Now, if 

 we have, by what precedes. 





Bi 



it is therefore necessary to find (o^E {A2, Bz). 



Now, 



w, {A,B,) = - A,-, 

 wi iA,B„) = - A,Au 

 ^ (.), (A.B^) = A^Ai - Ai- ; 



hence. 



u,E{A,,B2) = - 2 {A,B,)A,A. - {A^Bo) {A.A^ - ^.=) + {A^B:) A,A, ; 

 and consequently we find 

 (23) E(B,, A.) = BMo' -B,\-2 {A^Bo) A,A, - {A,B,) {A,A, - A,') 



+ (J,B,)(^„^,) + A.[{A,B,y - (A,B„)(AzBO]. 

 To find E{A3,B3) we employ the formula 



E{A3,B,) = - B^c ^' E{B3,A,), 



where 



d 



d 



a^a dAi 



d 

 dAi 



d 

 dAi 





and we now proceed to find the value of BiC Ar, where r has any 



integer value from to 2. 



We have B,e ^^ ' A,. = bJa,- - Az^ 



[A^Br), 



also 





c " {AyBs) - {A,Bs), r and s being any two integers 

 since QioiArB,) vanishes identically. 

 We have consequently 



I B^e ^' "Az = - (.Ja^.), 



(24) 



B,e ^' J, = - (^3^.), 



--n 

 B,i ^' 'a, = - (^3^0), 



--no 

 c~^'^ \a.B\) = (A,.B.,). 



