STOUFFER: INVARIANTS AND COVARIANTS. 63 



The transformations 

 _ 611 &12 



\i' Si' 



(18)^ &21 622 



\i' \i' 

 i = i(x). 



which leave B unchanged in form may be considered as consisting 

 of the transformation 



(19) < "^ D = hn 0-22 — O12O21 9^ 0, 



\ z = hiY + h22Z, 



in which $ = x, and of the transformations 

 ^ _ 1 



y = ~=^y, 



\r 



(20)^ _ 1 

 \r 



in which 611 = 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) DQik = r i Bji6it9ii, {i, k = 1,2). 



1=1 j=i 



If the transformation (19) is made infinitesimal by putting 

 6ii = 1 + c-i of and h.^ = c-j oi, (J ^ j)^ where c^ are arbitrary con- 

 stants and 'H an infinitesimal, the infinitesimal transformations of 



^ik are found from ( 21 ) to be 

 _ 2 _ _ 



(22) Sq-^= I {e-^^q-^-<f^q-^^)dt, {i,k=l,2).. 



j=i 



In accordance with the Lie theory the desired functions must 

 satisfy the system of partial differential equations. 



(23) Ursf^ '^ Cqir4^-qsi4^) = 0, (r,s = 1,2). 



1 = 1 "91s " Qrl 



Between these four equations there are the two relations 

 ^24) Un + U22 = 0, 



