66 THE UNIVERSITY SCIENCE BULLETIN. 



The system of differential equations for the semi-canonical 

 form of the semi-covariants is the same as the system for the 

 semi-canonical form of the seminvariants except that each equa- 

 tion contains more terms and there are four more variables. The 

 relations (24) and (25) both cease to hold so that there are three 

 semi-covariants or four relative semi-covariants. 



Equations (19) and (21) show that the expressions qny-{- 

 qn z and q2\ y + Q'22 z are transformed cogrediently with y and z , 

 respectively. The same is of course true of q'ny -\- q'nz 3,nd 

 q'21 y + q'22Z, respectively. It follows at once that the three ex- 

 pressions 



f C = (qny -\-q12z) z - {q2iij-\-q22Z)y, 



(32)^ E= (q'ny +~q'i2z)z- (q'2iy -hq'22z)y, 



\^0 = {qny + qv2z)z'- {q2\ y + q22 z) y' , 



are independent relative semi-covariants. A comparison of (19) 

 and (30) with (27) and (28) shows that the semi-covariants ( 31 ) 

 and (32) can be expressed in terms of the original variables and 



coefficients if y is replaced hy y, z hy z, y' hy :> and z' hy rr at 



the same time that q,\, and 9'ik are replaced by ?/ik and V\^, re- 

 spectively. Thus we have 

 f P = y<7 - zr, 



C = (uny + ui2z) z — {U'2\y + U22Z) y, 



E = {vny + V12Z) z - {V2\y + V22z)y, 



= {uny + Ui2z)t — {U2\y + U22Z) i> . 



By the same argument as in the case of seminvariants these 

 four semi-covariants are known to form a complete system 

 for (A). 



4. The Canonical Form and the Invariants. 



We shall now proceed to find those functions of the seminvari- 

 ants in their semi-canonical form which remain unchanged except 



for a factor Tfyn by the transformation (20). We shall thus 



obtain the functions of the coefficients of (B) and their deriva- 

 tives which remain unchanged by (18), except for the factor 



1 



(33: 



"^ 



