DR THOMAS MUiR 



a \ 



/'I 



'•] 



a l 



A 



7i 



rto 



h -l 



c 2 



a., 



& 



y-.' 



«3 



>>, 



c 3 



a 3 



ft 



ys 





'h 



«1 



^1 



'■>7i 



^ 



^2 



Vo 



s 2 



r| 2 



'•% 



^2 



4 % 4 



't3 



»"% r 4 » 



or D say, 



where the £'s, >/'s, ^'s are any quantities whatever. Performing the operations 



x colj + ij col 2 + z colg , x col 4 + ?/ col 5 + z col 6 , 



we obtain a determinant equal to x-D whose fourth column is r times its first, and 

 which therefore vanishes. D is thus seen to be equal to for all values of the 

 £'s, >/s, £'s. But D is clearly equal to 



a.. 



\ 



C 2 



a.-, 



a 3 



h 



H 



a^ 



ii 



Vi 



tl 





$o 



r}. 



C 2 





(s 



% 



I, 





rn 2 /3 2 - rb 2 

 r>u B, - rb a 



y., - r<\ 2 

 y-s ~ ' r 3 



that is 

 hence 



— a cubic 

 x,y,z. 



- l £i%£s I • I «h - • '«i Pi - r,i -2 y 3 



re, 



| aj - raj /J 2 - r& 2 y 3 - ?-Cg |=0, 



equation for the determination of r in terms of the original coefficients of 

 The general theorem manifestly is : — If we have given 



a 1 x l + b-^x.-, + 



<V'l + P\ X 2 + 



+ ^n 



a 2 x x + 



b 2 x. 2 + . 



. 4- 



lefS n 



a. 1 .r l + 



P-2 X 2 + • 



. + 



"^n 



a„x } + 



b n x 2 + . 



. + 



' ii '" a. 



then 



a n X x + jS 7 fl\ 2 + 



+ K,x„ 



\< h - ra, b 2 - r(3. 2 ... l n - r\ n \ = 0. (I) 



(3) If the number of equivalent fractions be one more than the number of un- 

 knowns, — as, of course, must be the case if elimination is to be expected, — it will be 

 possible to deduce as many results of the type (I) as there are fractions. Thus if l'/r 

 in the example just dealt with be also equal to 



"a x + Piii + y* z ' 

 we have oniy to leave out in succession the 4th, 1st, 2nd, 3rd fractions, and we obtain 



I a i - m i l j -i ~ rh -> ys - rc 3 1 = ° 



! a 2 - ra -i Pa ~ rh 3 Ji - rc i I = ° 

 a 3 - ra 3 B 4 - rb i 7l - rc x | = 



| a 4 - ra i /3, - rb x y 2 - rc 2 \ = 



