422 Mr. S. Lupton on the Reduction of 



Reduce any number of equations to the form 



u x — bx l + cx 2 2 + dx 3 . 



Multiply each equation by the coefficient of b in it and add all 

 the equations ; the resulting " normal equation " when solved 

 gives the most probable value for b. Proceed in the same 

 manner for c and d. Writing X 2 for x x 2 + x 2 + x 3 + . . . -f x n 2 , 

 we have : — 



Ui%i 4- u 2 x 2 + + u n x n = X 2 Z> + X 3 c + X 4 c?, 



u x x? + u 2 x 2 2 + + u n x n 2 = X 3 & + X 4 c + X 5 d, 



u Y x? + u 2 x 2 z + + u n x n d = X 4 Z> + X 5 c + X 6 d. 



The solution of these equations is generally performed by 

 Gauss's method, but at the best is very long and tedious. 



The following method is much more simple and convenient, 

 and affords a test of the real value of the equation found. 



Subtract each equation from the one below it and divide 

 by x 2 — x v and the similar terms 



\fi 



§Ui — — = b + x 2 + x x c + x 2 + # 2 #| + Xi d, 



X 2 X\ 



\qi 



Bu 2 = - — — = b + x 3 + x 2 c + x z 2 + x 3 x 2 + x 2 d, 



x 3 ~~ x 2 



AW 3 



Su B = = b + x 4: + x 3 c + x 4 2 + x 4t x 3 + x 3 2 d. 



x^ x 3 



Subtract again and divide by x 3 — x x and similar terms, 



S2 A ^l , J 



0^1= = — =-=c + x 3 + x 2 + x x d, 



X 2 — Xi X 3 — Xi 



A 2 u 2 



Pu 2 = = = c + x 4 + x 3 + x 2 d. 



x 3 x 2 x^, ~~~ x 2 



Subtract again and divide by x 4 — x l : 



d 3 u 1 = =d. 



X 2 "■ — X± X 3 ~~~ X-i X a "~~ x± 



Substituting successively in these equations 

 d=S 3 it l7 



C= B 2 Ui — X 3 + X 2 + Xi B 3 u ly 



b = Su 1 — x^ + Xi h 2 u x -f x 3 x 2 -f x 3 x x + x 2 x x S 3 Mj, 

 a = u , — bx x — tXx — dxi. 



