14— THEOREMS REGARDING AGGREGATES 

 DETERMINANTS AND PFAFFIANS. 



By Thomas Muir, LL.D. 



OF 



I. Of several familiar results, each of which may be made the 

 starting point for an exposition of the theorems here to be dealt 

 Avith, the least likelv, in view of the title, is the identity 





^^) 



where U is anv homogeneous function of .v, \', .z of the second 

 degree. Nevertheless, if for U we take 



ax' + hy'^ + cz- + 2fyz + 2ozx + 2lixy 



or. as it is best written in order to show its discriminant, 



X y z 



it will be found that each of the terms on the right of (A) may be 

 expressed as a determinant : the unlikelihood is thus at once 

 considerably diminished. The identity then becomes 



= \ a z —y 

 ■^ h . X 



y \i^ -X . 



y -X 



Avhere. be it observed, each of the three determinants is formable 

 by taking two columns from the zero-axial skew determinant 



~ -y 



-z . X 



y —X 



and one from the discriminant of the quadric. 



2. // P be any n-liiie dcteniiinaiit, O a zero-axial skew detenninanl 

 •of the same order, and Ar a determinant formed by replacing the r^^ 

 jrolnmn of O bv the r^^ cohimn ofV, then, ivhen n is odd. 



:S4.= 



^1. — /Z2, /^s. 



-A^4, 





