STOUFFER: INVARIANTS AND COVARIANTS. 63 



The transformations 

 '^ 611 bi2 



(18)^ 



621 622 



which leave B unchanged in form may be considered as consisting 

 of the transformation 



(19) M ,, , ' Z) = &11622- 612&21 ?^ 0, 



I z = 021Y -i- O22Z, 



in which i = x, and of the transformations 

 ^ ^ 1 



(20) -<i _ 1 



V ^ = Hx), 



in which hn = 622 = 1 and 612 = 621 = 0. 



2. The Seminvariants. 



Let us first find those functions of the coefficients of ( B ) and 

 their derivatives which remain unchanged in value by the trans- 

 formation ( 19) . Equations ( 17) show that ( 19) converts q,^ into 

 Qik where 



(21) DQ,k= ^' '^^ By,b^q,„ {i,k= 1,2). 



1=1 i=i 



If the transformation (19) is made infinitesimal by putting 

 6ii = 1 + fn ''^>t and b\i = fij <U,{i 9^j), where v'lj are arbitrary con- 

 stants and H an infinitesimal, the infinitesimal transformations of 



gu, are found from (21) to be 

 _ 2 _ _ 



(22 ) gu, = - ( f]k Qvi - U 9jk) '^^, (i , A; = 1, 2) . 



i = i 



In accordance with the Lie theory the desired functions must 

 satisfy the system of partial differential equations. 



(23) Ursf^ H~qir-^-qsi~^) = 0, (r,s = l,2). 



Between these four equations there are the two relations 

 ^24) [7n+f/22 = 0, 



