STOUFFER: INVARIANTS AND CO VARIANTS. 61 



The substitution from (4) into (3; now gives 



,_, \ «]1 y" + «12 Z" -(- (Wii an + Wi2 «21 ) V + (Wll «12 + Wl2 «22 ) I = 0, 



(5) j - - 



/ «21 2/" + '^22 2" + (W21 '^11 + 2^22 '^21 ) Z/ + ( W21 «12 + W22 «22 )Z = 0, 



where* 



2 



(6) i/ik = gik - p'ik - 2 Pij pjk, ( 2, ^- = 1, 2 ) . 



i = i 



The system (5) may be put into the form 



(■o ; < _ 



(2" + g2i2/ + g22z = 0, 



if we write 



■1 ■■) 



(7) A gik = -"■ -"' hi «ik %, (i, A- = 1, 2) , 



i=ii=i 



where -^i is the algebraic minor -/j; in the determinant of the trans- 

 formation (1). WilczjTiski calls {B ) the semi-canonical form of 

 the system ( A ) . 



The differentiation of equations { 7 ) gives 



9 2 



(8) A?ik =_2 i;r-iji'^ikM'ji + -Jji«'ik% + -■''ji'^'ikWji]- 



A'^ik, ihk = 1, 2). 

 By the use of i,4 ) we find 



2 2 2 



i = l j = l m = l 



A' = - (pll + P22)A, 



whence it follows at once that 



(9) A g'ik = '- '- --'ji ^^ik%, {i, A: - 1, 2), 



1=1 j=i 



where 



2 



(10) rik = z/'ik+ 2 (pij«3k-P:kWij^, (/, A- = l,2). 



j = i 



It follows without calculation that 



( 11 ) A ?ik = ^' 2 .-Iji a,k ii'ji, n", A- = 1, 2 ) , 



i=ii=i 



where 



2 



(12) Wik = i-'ik+ - (Pij?'jk -Pjk^'ij), (/, A- = 1, 2). 



j=i 



♦The expression here used for Ujk differs in sign as well as in numerical coefficients from that 

 used by WilczjTiski . 



