THOMAS MCJIR ON BIPARTITE FUNCTIONS. 



470 



\ x \Vi z z\ > K A 7s I > \ a i b 2 c z\ ; 



product of 

 hence, putting 



I a i h C B I = Al . I a l & 7 3 I = A2 . K #2 % 



we may write the above identity in the form 



= As> 



w^ w a r x 



^_ 





m x 7i T ?*j 



4 







A 3 



Si 



S 2 

 S 3 



A 3 A 2 A 1 



% 



and the like holds when there is any number of square arrays A i , A 2 > A 3 , 



A 4 , .... 



As another instance, the theorem of § 27 may be taken, which may now 

 stand thus — 



a l a. 2 a s 



A x 



A, 



A + 



A, I x 1 y x z x ~ A, • Aj 



a, y, 2, 



A 4 -A, 



but in the case of gwjf theorem we have given regarding the condensation of 

 bipartites a like simplification of expression is possible. The only points 

 requiring attention are — (1) that, in forming the determinant of any square array, 

 we must take for the first row that line of the square array which is contiguous 

 with the bar separating it from the previous square array ; (2) the initial and 

 final lines of a bipartite are to be looked on as lines of a determinant whose 

 other elements are all zeros. 



44. Qualities are expressible as bipartites. 



(a b c d$x 7/f is 



x y 



X 2 



Thus the binary cubic 



a b 

 b c 



and the ternary quadric 



2xj/ 



ax* + by' 1 + cz 1 + 2dxy + 2exz + %fu z 



is 



x y 



a d ex 

 d b fly 



e / c z . 



45. A notable characteristic of the bipartite expression for a quadric is that 

 it brings into evidence the discriminant of the quadric — the discriminant, in 

 fact, being the determinant of the square array of the bipartite. This suggests 



