OF INTERPOLATION AND EXTERMINATION. 343 

 (a h — a h \ x-\- (h c — h c \ z:=: Ah — A b ; and 



^13 31' ^31 IS' 13 31 



from these two results we obtain (a b — a b ) (be — 



^12 2 J ' ^ 3 1 



be) X — (a b — a b \ (b c — 6 c \ x=. (A b — A b \ 



1 S'^ ^13 3 1^ ^ 2 1 12/ ^12 2 i/ 



(b c — b c \ — /A b — A b \ (b c — b c \ : whence, by 



^31 l3>' Vl3 3i/V21 12'' 



actual multiplication, we have abbe, or Abbe, marked thus, 

 (l,2,3,l~l,2,l,3-2,l,3,l+2,l,l,3)-(l,3,2,l-l,3,l,2- 

 3,1,2,1+3,1,1,2), or since abbe=iahbc, (—1,2,1,3 



1 2 3 1 1 3 2 1 



—2,1,3,1+2,1,1,3)— (-1,3,1,2-3,1,2,1 +3,1,1,2) which is 



divisible by — b , and may therefore be reduced to (1,2,3+ 



2,3,1-2,1 ,a-l, 3,2 - 3,2,1 + 3,1,2) = I, (2,3-3,2)-2, 

 (1,3 — 3,1) +3, (1,2 — 2,1). And in the case of 4 equations, 

 the analogy leads us to the value a=& (e d — c d \ — b 



2^34 43/ a 



(e d — c d \-\-b (c d — c d \: but in all such cases, a 



A24 42>' 4W3 32/ 



numerical computation has the advantage in conciseness, 

 because the sums or differences of two numbers are as 

 easily multiplied as the numbers themselves. 



The process may also be represented in a symmetrical 



manner by calling the second series of equations a! x 



1 



+ 6' y + c' z +...■=: A' , a' a:+...=J.' , the third series 



11 12 2 



a"a: + . , .-=.A" ,.,, until at last x is left alone on one side : 

 1 1 



a a c e a a 



and so forth. 



Scholium. We may take for an example the equation 

 y=o + 6x + cx^ + dx' + «ar*, to be determined from five va- 



