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 + 

 gi2 2 and q2iy + q-i-iz are transformed cogrediently with y and z, 

 respectively. The same is of course true of q'ny -\- q'uz and 

 q'2\y +~q'22Z, respectively. It follows at once that the three ex- 

 pressions 



( C ^ Cqny-^qnz) z- {q2i V -\- q22 z) y , 

 (32)^ E = (q'ny -{-'q'v2z)z- {q'2iy i-q'22z)y, 

 1^ = Cqny+^i2Z)z'- {q2\V + q22Z)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 by y, z by z,y' hy [> and 2' by t at 

 the same time that q,^ and g'lk are replaced by Wik and %, re- 

 spectively. Thus we have 

 f P = y<7 -zp, 



C = {uny + ui2z)z- {U2\y -\-U22z)y, 



E = {vny + vuz) z - {V2iy -}- V22z)y, 



= (uny + Mi2 2)'5- - {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 \t'\m 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 



Wr ' 



(33)^ 



