STOUFFER: INVARIANTS AND COVARIANTS. 67 



Equation ( 17 ) shows that ( 20 ) converts ( B ) into a new system 

 whose coefficients Qik are given by the equations 



fQn= 7w i^r-l^'^q.^, {i= 1,2), 

 Qik = TTTy^ 9ik, (t,k = 1,2; I 9^ k). 



V 



We notice that 



Qu + Q22 = j^Ty ( 2 '/"'^ - 'z ' + gil + qi2), 



so that Qii + Q12 = 0, provided that 



(35) //. = r/ - }, t/-' = g-n + g22. • 



From equations (34) we have at once, if (35) is satisfied, 



(36) 



^^ 



Qii= Y^iq-u-lD, {i = l, 2), 

 U ) 



Qik=7|W9ik, {i,k = 1,2; 19^ k). 



whence 



(37)-<^ 



V 



"" Q'u = T^ [q'u -\r -2r, {q^ -hi)], 



Vt ) 

 Q"n= JJTy [?,i-U"+/— 27g,-5^(g'„-U') + 



^vCqu-^D], 

 Q'ik = -|^(g'.k-2-/;9.k), {i,k = 1,2; i 9^ k) , 



Q\ = -^ (?ik - 2 7 gik - 5 rg',k + 5 V^ 9:k) • 

 Let us now assume that (B) has been converted into 



j ? + Q2li + Q22^- 0, 



where Q,k have the values (36) so that Qn + Q22 = 0. The sys- 

 tem ( D ) is called the canonical form of ( A ) . 



If the seminvariants for (D) corresponding to I , J , K, L for 

 (B) are denoted by /i, Ji, Ki, L\, respectively, equations (37) 

 show that 



