38 DR MUIR ON THE ELIMINANT OF A SET OF GENERAL TERNARY QUADRICS. 



the first part of which is established by multiplying the determinant on the left by 

 | T73& I in the form 



Vi P s • 



>h Pi ■ 



>/2 & 1 



and then removing the same factor from the product; and the others by making a 

 cyclical change in the rows of the original determinant and using the part already 

 proved. 



Thus, putting a, & y, £ , h J=a, b, c,f, g, h, we have 

 I I «i/ 2 I I « 2 / 3 i I a Ji 



I h i9* I I \g A I I \y x 



C A 



c 2 h 3 



cj h 



= 59' + 67 = 67' + 4'8 = 48' + 5'9 ; 

 putting a, ft y, £ v , £=a, b, c, g, h,f, we have 



= 111 -48' = 212-59' = 310-67'; 

 and putting a, ft y, £ v , f = a, 6, c, ft, /, #, we have 



= 4'8 + 3 10 = 5'9 + 1 IT - 6'7 + 2 12 . 



a i92 I I a oJ S I I «3#1 

 & A I I 6 2^3 I I h h l 

 C l/ 2 I I C 2 / 3 I I c zfi 



a 2^3 I I a 3^1 



a 1 A 2 



& i/ 2 I I ^2/3 I I 63/1 

 °i9% I I H9% I I H9x 



What is still more interesting is the fact that, when £ 77, £ are made equal to ft y, a, 

 the above theorem in compound determinants degenerates into the theorem regarding 

 the adjugate ; and that we thus obtain 



a A I I a $z I I a A 



¥2 



c x a 2 



& 2 C 3 



h c l 



= 00, and 



I flffs I I / 2 #3 I I fsffi I 



i 9 A I I # A I I 9 A I 



= 0'0' 



The diversity in the extraneous factors is thus seen to disappear entirely, the results 

 being 



y, = I 



aJ) a 



b 9 c 



2^3 



3«l I I • E = y 5 , 



y-2 = I \fi92 1 1 Ms 1 1 Vi 1 l ,E ' 



73 = -| |«AI I \A I I «tfl I l-E, 

 V4 = -| fl^ 8 1 \b 2 h s I J oj/j I |-E. 



