STOUFFER: INVARIANTS AND COVARIANTS. 



71 



There is another expression for an invariant which is easily 



obtained and which is of geometrical interest. From equation 



(21) we easily deduce the equations 



D (Qn — Q22 ) = ( 611 622 + 612 621 ) {qn — q22) + 2 621 622 912 - 2 612 611 921 , 

 DQ12 = 612622 {qn — 922) + 622-q'i2 — 6i2"-^ 921, 



D Q21 = - 621 611 (g-ii — q-22)— hii^qn + 611- q2\, 



and exactly similar equations invohing derivatives of any order. 



Thus we know at once that the determinant 



(46) 



is the semi-canonical form of a seminvariant. Furthermore 

 equations (39) and (40) show that it is the semi-canonical form 

 of an invariant. The expression in terais of the original coeffi- 

 cients for this invariant is 



(47) 



5. The Covariants. 



Let us now return to the semi-canonical form of the semi-co- 

 variants and assume that they have been written down for 

 equations (D). If they are denoted by Pi, Ci, Bi, Oi, equations 

 (39) and (40) show that their values for the equations obtained 

 by transforming (D) by (20) with ^ = are as follows: 



Pi = Pi, Ci = -^C,, 



(? ) {^ )' 



Therefore four relative covariants in their canonical form are 



Pi, Ci, £"1 + 401, 2JiEi-CiJ'i. 

 By converting these expressions into the original coefficients and 

 variables we find the complete system of covariants for (A) tobe 

 P, C,Cs = E + 4(0-UP) =E + 2N,C: = 2eiE-0\C. 



