342 APPENDIX. B. 



ing obtained from the original coefficients, by 

 exterminating all the unknown quantities, ex- 

 cept ^5 in succession. 



For example, if there are two equations between x and 



b A —b A 

 y, we have a=:b and ^mb , and xzz^—l — ^ — 2.: if 

 2 , a — 6 a 



there are three, between x, y, and z, we have a-=zb c 



2 1 12 



2 3 



h c , ^=6 c — b c and y—b c —b c . It will readily' 



32 1331 l22i 



appear in all cases, that at every step of the process of 

 extermination, the quantities a and A are multiplied or 

 divided by the same factors, so that when all the other 

 quantities are exterminated, the factorofx, which remains, 

 must contain all the as, with the same factors as belong to 

 the ^s on the opposite side of the equation. Thus, for 

 two equations, a x-\-b y^=-A ,and a x-\-b y=-A , multi- 



111 2 2 2 



plying- the first by b and the second by b , and taking 



2 1 



their difference, we have a b x — a b xzzA b — A b : or 



12 21 12 21 



a a A A 



dividing by b and b respectively, -i x — -2. a:=— -^ — —1, 

 13 b b b 



1 2 12 



which obviously leads to the same result. For three equa- 

 tions 



a x-\-b y-\-c z—A ; 

 111 1 



a x-\-b y + c z—A ; 



2 2 2 2 



a x-^-b^y + c zzzA ; we have first a b x-\- .. +6 c zzr 



3 3 3 3 12 2 1 



A b , a b ar + ..+6 c z=:A b , whence (a b — a b \x 



1221 12 21 ^12 21'' 



+ /6 c — b c \zzzA b — A b ; and in the same manner 



\ 2 1 12/ 12 21 



