STOUFFER: INVARIANTS AND COVARIANTS. 65 



system of seminvariants for the system ( A ) , To show these 

 facts let us suppose that we have two systems of form ( A ) which 

 are equivalent under a transformation of form ( 1 ) . Each of 

 these systems may be reduced to a semi -canonical form and these 

 must be equivalent under a transformation of form (19). A 



seminvariant expression, qn + g'22, say, formed for these two 

 semi-canonical forms must be equal and each is equal to the ex- 

 pression Mil + U22 = / formed for its corresponding original sys- 

 tem. Therefore the two expressions for I are equal and / must 

 be a seminvariant. The same reasoning applies to the other 

 expressions (26). That we have a complete system of sem- 

 invariants is obvious from the fact that every seminvariant of 

 (A) must have a semi-canonical form which remains unchanged 

 by transformations which leave the semi-canonical form invariant. 



3. The Semi-Covariants. 



We shall now find the semi-co variants of (A) by finding first 

 the semi-canonical form of these semi-covariants. The trans- 

 formation ( 1 ) when solved for y and z has the form 



(27) S A^ = '^22^ - '/-iL's;, 

 I Az = — a2iy -^ o.nZ. 



When the coefficients of this transformation are subjected to the 

 conditions (4) we find 



(28)1 A^' = -2/'-'/i2'^. 

 I Az' = — '«i,"+ «ii'7. 



where 



(29) ," = y' -\- pny -\- p\2Z, 't = z' -\- pny -\- p 22 z. 



Evidently semi-covariants need contain no higher derivatives of 

 y and z than the first. 



The semi-canonical form of the semi-covariants will be found 

 by subjecting (B) to the transformation (19). Since the coeffi- 

 cients in (19) are constants 



.3Q. j y' = bnY'-hbi2Z\ 

 I z' = h2xY' -\-h22Z' , 



and it follows at once that 

 (31) P = yz'-'y'~z 

 is a semi-covariant. 



5 — Sci. Bui. — 860 



