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 undei' 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 

 semin variant expression, qn + q22, say, formed for these two 

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

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

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



\Ye shall now find the semi-covariants 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)) A^ = «22 2/ - «i2 2;, 



I A 2 = — «2i2/ + unz. 

 When the coefficients of this transformation are subjected to the 

 conditions (4) we find 



A ?/' = «22/' — «12'^, 



(28) , 



A ^' = — '-'21/'+ «11 '^ 



where 



(29) r = y' -\-puij -\-pr2Z, '7 = z' -h P2iy -\-p22z. 

 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 (jB) to the transformation (19). Since the coeffi- 

 cients in (19) are constants 



(30)) y_' = b^^y' + ^-^'^ 



^ I Z' = h2xY' + h22Z' , 



and it follows at once that 

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

 is a semi-covariant. 



5 — Sci. Bui. — 860 



