STOUFFER: INVARIANTS AND COVARIANTS. 67 



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

 whose coefficients Qa^ are given by the equations 



(34)<; J _ 



I Qik = 7|7jo 9ik, {i,k = 1,2; I 9^ k). 

 We notice that 



so that Qii + Q22 = 0. provided that 



(35) ,'1 = r/ — \ r/- = qii -f q.>2. 



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

 { Qu= -|^(gii-i/), {i = l, 2), 



(36)^ ^ "1 - 



I Qik = -|7Ty9ik, (i,k = 1,2; i ^k), 



whence 



Q'i. = T|^ k'ii - U' - 2 >; (gli - U) ], 



Q"i. = ^ [9"ii- U" + /■-' - 2/g, - 5/, (g'ii - U') + 



Q'ik = j^ (9'ik - 2^ gik), (I, k = 1, 2; y ^ ^•), 



, Q"ik = -re7yi (9"ik- 21 q^- 5r~q'^^-\- 5v q^) . 



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

 y" + Qn~y + Q12Z = 0, 



r 



z" + Qny + Q-^2Z^ 0, 



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

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



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

 (B) are denoted by h, J\, K\, L\, respectively, equations (37) 

 show that 



