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) = {bu h22-\-hv2b2i) {qn — 922) +2621622912- 2612611921, 

 DQ12 = 612622(911 — 922) + 622^912 — 612'^ 921, 



D Q21 = - 621 611 (911 - 922) - 62i''^ 912 + 611'^ 921 , 



and exactly similar equations involving derivatives of any order. 



Thus we know at once that the determinant 



(46) 



911 — 922 912 921 

 9' 11 — 9 '22 9 '12 9 '21 

 q"n-q"22 ^'vi ^2\ 



is the semi-canonical form of a semin variant. Furthermore 

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

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

 cients for this invariant is 



(47) 



6. 



un — U22 



I'W — ?'22 



WW — W22 



1(12 



ri2 



IC\2 



M2I 



1-21 



W2I 



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, fii, Oi, equations 

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

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



Pi = Pi, 



(r) 



Ci = 



{El 



r 



Ci, 



2-/;Ci),Oi = 7^(Oi + ir;Ci) 



(? )" 



Therefore four relative covariants in their canonical form are 



Pi, Ci,£;i + 40i, 2JiEi-CiJ'i. 

 By converting these expressions into the original coefficients and 

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

 P, C,C3 = E + 4(0-i/P) =E-{-2N,Ci = 20iE-e'iC. 



