DR THOMAS MUIR 



b x y + c x z a x X + fa y x 



b 2 y + e 2 z a 2 x + fa y 2 



b 3 y + C a 2 a 3 x + fa y 3 



b$ + C 4 Z a 4 X + fa y, 



a x x + b t y + CjZ 04 f3 x y 1 

 a. 2 x + b 2 y + c 2 z a 2 ji 2 y 2 

 a 3 x + b 3 y + c 3 z a 3 (3 3 y s 



a i x + \y + c 4 z « 4 & y 4 



a x x + b x y Cj a x fa + y x z 



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



a 3 x + b s y c 3 a 3 fa + y 3 z 



a 4 x + b 4 y c 4 a 4 f3±y + y 4 2 



a i ^1 c i a i x + fi\!J + 7i z 

 a, 6 2 c 2 a 2 x + /3 2 y + y 2 z 

 a 3 b 3 c 3 a 3 x + fa + y 3 z 



C 4 a 4 X + fay + y 4 2 



= 



= o, 



a 4 b \ 



= 



The first and third of these are linear in x , y , z; in the second the two determinants 

 are quadrics, the facients being xy , // 2 , zx , yz ; but as the coefficient of zx is | a 1 c 2 a 3 y i j 

 in both determinants, it is possible to remove the factor y, thus making the equation 

 linear also. We consequently have 



a i a 2&y 4 1 x + 

 a A a sy4 i ) 



X + 



- | a x c 2 a 3 p 4 



i 



a x b 2 c 3 a i | x + 



\b 1 c 2 a 3 /i i \ J ' - l&jCjjOgyJ ) 



I a \h c aPi \y + I a A c 3T4 i z = ° ' 



and thus have solved the problem set ourselves. 



In eliminating x , y , z from these, we obtain a result agreeing with that of § 4, and so 

 learn that the set of linear equations equivalent to the set of quadrics specified in the 

 enunciations of the theorems of §§ 3, 4 has for its coefficients the elements of the 

 conjugate of the eliminant obtained in the latter paragraph. (V) 



The law of formation of the vanishing determinants which originate the said set of 

 linear equations will make its appearance if we take an additional case, say the case 

 where the given equations are 



a m x + b m y + c m z + d m w 1 



a m 3 + fi m y + y m z + S m «0 r 



(m = l, 2, 3, 4, 5). 



Here the first linear equation is 



a x x + b x y + c x z + d x w 



Pi Yi 8 i 



a 5 x + b 6 y + c s z + d b w a- /3 S y 6 



= 0, 



where all the 4 terms of the numerator are kept together, and all the 4 terms of the 

 denominator are separated ; the second is 



