STOUFFER: INVARIANTS AND COVARIANTS. 61 



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



\ «n y" + '^12 z" + (wn 'in + W12 '-'21 ) ?/ + (wii '-'12 + W12 "22) z = 0, 



(5) < _ _ _ _ 



( «21 ?/" + '-'22 2" + ( W21 'ni + U22 "-IX ) y -\- { W21 ai2 + U22 «22 )2 = 0, 



where* 



2 



(6) Wik = gik -p'ik- SpijPjk, (/, fc = l, 2). 



j = i 



The system (5) may be put into the form 



,^\u"+'Qn~y-\-qi2Z= 0, 



{^)i - _ _ - _ 



( Z" + 921 2/ + 922 2=0, 



if we write 



_ 2 2 



(7) A g.k = - - -hi ''Ik Mil, (/, A; = 1, 2) , 



i=ii=i 



where -^ji is the algebraic minor '>(ji in the determinant of the trans- 

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

 the system ( A ) . 



The differentiation of equations ( 7 ) gives 



_ 22 



(8) Ag'ik= 2 - [•''ji "11.^^1+ .^ji'^'ikWji + --I 'ji'^ikMjil- 



1=1 j=i _ 



A'9ik, {i,k = 1, 2). 

 By the use of (4) we find 



22 2 



- -^'jiWji = - --Iji [- iVn + P22) Wji + - PjmWu,i], 



j=l j=l m=l 



A' = - (Pll + P22)A, 



whence it follows at once that 



(9) A9'ik= "- ^'•-'ji"iki'ji, (z, A: = l,2), 



1=1 j=i 



where 



2 



(10) ?^ik = w'ik+ 2 (pijWjk- PjkWij), (2,A; = 1,2). 



j = i 



It follows without calculation that 



( 11) A g"ik = ^' 2 Ay, '^ikWji, (i, k = l,2), 



i=ii=i 



where 



2 



(12) Wik = w'ik+ - (Pij^jk-Pjk^'ij), (i, A: = l,2). 



j=i 



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

 used by Wilczynski. 



