ON ELIMINATION IN THE CASE OF EQUALITY OF FRACTIONS. 5 



where D 1 = \ a x b 2 c z . . . l n a n+1 \, D 2 = | ai6 2 c 3 . . . l n fi n+ i \ , . . : and where D' r indi- 

 cates that any one of the italic letters of D,. except the r th lias been replaced by the 

 corresponding Greek letter, D" r that any two letters except the r th have been similarly 

 treated, and so on. 



As before, it has to be noted that by changing the D's of (III) into A 's (viz. 

 A ! = | ai/3 2 . . . \j<x n+1 1 , A 2 = I «i/3 2 . . . \ib n+ i j , . . . ) we merely reverse the order 

 of the columns. 



(5) The eliminant just obtained being different in form from that reached in § 3, we 

 are thus furnished with a very interesting identity in determinants, the establishing of 

 which is well deserving of attention.* (IV) 



The simplest case of it is 



a A I I a A I + I a \fi2 I ! a \P-2 



a 2 b 3 | | a 2 b 3 + | a 2 fS 3 | | a 2 /^ 3 



a 3 h l i I a 3 h \ I + I "-A I I a -A 

 Here the operations 



a-fi.-,^ | | aj/3 a 3 | 



a 3 -roWj 4- a 1 -row 2 + a.yVO\v. i , 



I a 3 Pl ! • I0W 2 ~ I "llPs I • r0W 3 » 



performed on the three-line determinant produce t 



Ojbfy 



I «iA> a 3 ! 



<*3 I a A#S I - Ai I a A a 3 I a 3 I a AAi I - ^3 ! a i/ 8 2 a 3 



a s/ 3 i 



a 3 ! a 3/ 3 l 



from which the two-line determinant readily evolves after the performance of the 



operation 



row 2 + /3 3 • row j . 



(6) 111 a previous communication to the Society attention has been drawn to the 

 importance, when dealing with elimination in the case of a set of quadrics, of discovering 

 the corresponding set of linear equations. Let us seek, therefore, the set of linear 

 equations in x , y , z which corresponds to the set of quadrics in § 3, viz. 



a x x + h x y + c x z a 2 x + b 2 y + c 2 z 



ajX + fitf + y x Z a 2 X + j3 2 ij + y 2 2 



a^x + b A y + c 4 z 

 ap + ~/3 4 2/ + y 4 z 



The mode of reasoning followed in §§ 2, 4 makes clear that on account of the 

 existence of the given equations we have 



* This and a cognate identity are formally proved in the Messenger of Math., xxxv. pp. 118-122. 

 t Note the identity, 



I »22/3 - I I2I3 1 



I X\'Mz I • 5 i2/ 2 '?3 





