DR THOMAS MUIR 



°i h i 



ft 7l 



a i 



&2 



C 2 



a. 2 



A 



y-2 



a. 



*3 



C 3 



a 3 



& 



y 3 



a 4 



^4 



C 4 



a 4 



ft 



74 



6 



Vi 



Ci 



*i 



"7l 



< 



£2 



V2 



£ 2 



^2 



I'Tj., 



't 2 



Proceeding exactly as in § 2 we can show that it must vanish for all values of the 

 £'s , >/s , £'s : and since it is equal to 



h 



ra x ft - i\ y x - r,\ 



that is, 



a 2 



b 2 



C 2 



a, - 



ra 2 



& - r\ 



72 - 



- re, 



a 3 



h 



'•3 



o 3 - 



ra 3 



& ~ rb 3 



7s - 



- re. 



a i 



K 



C 4 



a 4 - 



ra 4 



& - rb 4 



74 - 



- re 



61 



Vi 



Ci 







• 







i% 



y-2 



U 



. 









, 



Vits 



a x a, - ra 2 /3 3 - r&, y 4 - rc 4 | + | &£, 1 



+ I £1% 



& 1 «2- m 2 / S 3- r& 3 74: — rC 4 I 



c i « 2 -ra 2 f3 s -rb 3 y 4 -rc 4 |, 



or 



I fl^ I • I a i a 2 #5 - ? ' /; 3 74 - J ' C 4 I + I £l4 I * I /j l a 2 - m 2 As 74 ~ V ' C 4 I 



+ I fl% I ■■ I C l u 2 - TO 2 A - rJ 3 74 I ' 



it follows that the cofactors here of | ^(^ | , \ £1^2 1 , ! ^2 1 must each be equal to zero. 

 This gives us the set of equations — 



I a 1 a 2 B 3 y i I - { I a^b^ J + I fl^Og/Jg^ I } r + \ a l a 2 b 3 c i | r 2 = 

 1 W^ I - { I V2/V4 ! + I 6 l«2i°374 I } r + I h<hP& i r2 = ° I 

 ' c l a 2^»y4 I - { ! f i a 2/3 3 74 ! + I C l a 2 & 374 I } r + I ^2^4 I »* = ° ' 



from which by elimination of r, r 2 we obtain 



« 1 a 2 /3 3 y 4 I j a 1 a 2 6 3 y 4 | + | a^fi^ \ \ a l a 2 b. i c i \ 



h l a 2p3Yi I I & l a 2A C 4 I + I V2A374 I I *l« 2 / 3 3 C 4 I 

 I C l a 2&$74 I I C i a 2/ 5 374 ! + ! t 'l a 2 6 374 I I C l«2 & 3 74 I 



The general theorem is : — The eliminant of the set of equations 



a x x x + h^x 2 + . . . + /,£„ 



= 



a l X 1 + ySj^o + . . • + \X„ 



Ct.jX-, ~T OcyXty "T . 



+ i 2 x tl 



a 2 X l + Pi X -2 + • • 



+ X 2 x n 



a n+l x i + n+ - i X. 2 + 



• ■ + ^)i+\ X n 



tt„ + 1 X! + f3„ +1 x 2 + . . . + X n+1 x n 



IS 



T> 1 2D', 2D", 

 D 9 2D' 2D" 9 



D„ 2D' fl 2D" n 



(III) 



