478 DR G. PLARR ON THE DETERMINATION OF 



These values give (as #, = S/<£ -1 #): 



^-ir- Y< ,-,.-gg + V'*^-* -|. 



« L r w- m J 



Hence 



fliMf, f = V. ?0iS/0 ~Vc-Y.i<p-' 1 iSj<pk . 



Developing Vicf>i and Vi^'H into their components parallel to/, A ; 



Vi0i = V.jk<f>i =jS>k(pi — k§j<j>i , 



Yicf>-H = Y.jk<f> - J i =yS&0 - H — kSj<f> ~H, 

 we get 



+ K - sy^y^fc + sj<p-HSj<i>k] . 



Now <j> and <£ -1 being self-conjugate, we replace Sk<f>i, Sk(j>~ l i, respectively by 



Si<pk, Si^k, 

 in the term affected byy, and 



by 



s%\ sty-y, s*0-y, s%\ 



in the term affected by X:. This gives 



mit '*l r i f = y [Si0&S/0 ~ *& — Sz'0 ~ l kSj<j)k] 



+k[— si^ys&^-y + si0 - yskfj]. 



The coefficients of y, & are respectively 



S . Yij V. - %^ = S.k<j>~ l k.(j>k 

 and 



s . v& V0 - y#- - s .y^ - y. #y . 



So that, with a change of sign {&k$~ l k.$k = — Sk^k^' 1 ^ &c), and introducing 

 the notation 



we have 



m^ 1 (^)=-yW 3 -7cW 2 . 



Let us further consider for any vector X : 



V. 0X0 - l \ = V. (S ^) (2aa»Sa\) 

 = a/3SaXS/3xg_fJ) 

 + 7 aS 7 XSax(^ — ^) 



+^s^s 7 xg_g. 



