373 
Roots of Equations. 
5. If the remainder Q =3 0, then, by the second observation, 
the introduction of cS in the division, produces no error in the 
conclusion ; and, if Q do not vanish, 0? will be found in every 
term of tbe third remainder, and may there be rejected ; and 
so on. Thus we obtain the conclusion, without any unneces- 
sary values of A, B, C, &c, or a, b, c, &c. 
6. If the highest indices of x , in the original quantities, be 
equal, it will only be necessary to multiply the terms of the 
dividend by A, which may be rejected after the second division. 
If the difference of the highest indices of x be m , the terms of 
the dividend must be multiplied by A OT +g the first quotient be- 
ing carried to m 4 - 1 terms. This quantity, A B + X , will enter 
every factor in. each term of the second remainder. 
7. If it be necessary to continue the division, let 
C* — B/ 3 . a 4- . A== mA? 
Doc, — ■ By. a -f- fiy — ocS . A = n A' 
E« — • BL a -I e . A=y> A- 
&c. 
and the third remainder is (yin — fin . m 4 - »*' — nip . «) x n ~* 
~\-(cim — fip.m-fnp — mq. a)x n ~^-\ ~{zm — fiq.m-\-nq — mr.u) 
x n ~ 6 -f &c. every term of which is divisible by of. The law 
of continuation is manifest 
prop. n. 
Two roots of an equation of 2 m dimensions may be found 
by the solution of an equation of 'Jn . 2m — 1 dimensions. 
Let x zm 4 - px ™- 1 4 - qx 2m ~ 2 -{- rx 2m ~ 3 4 “ & c - — 0 ; and, if 
possible, let v and % be so assumed that v % 3 and — v 4“ 
may be two roots of this equation ; then, 
