342 PHILOSOPHICAL TRANSACTIONS. [ANNO 1798. 



it gives us no insight into the subject. Dr. Waring's observations on the propo- 

 sition are extremely concise ; and, to common readers, it will still be a matter of 

 doubt, whether a quantity of any description whatever will, when substituted for x 

 in the expression x* — px* + qx 6 — . . . . + w -> cause the whole to vanish. 



In the investigation of the proof here offered, it became necessary to attend to 

 the method of finding the common measure of 2 algebraic expressions ; and to ob- 

 serve particularly, in what manner new values of the indeterminate quantities am 

 introduced; and how they may again be rejected. It appears, that these values are 

 necessary in the division; and, when they have been thus introduced, they enter 

 every term of the 2d remainder, from which they may be discarded. This circum- 

 stance enables us, not only to determine the nature of the roots of every equation, 

 but also affords a direct and easy method of reducing any number of equations to 

 one, and obtaining the final equation in its lowest terms. 



Pbop. 1. To find a common measure of the quantities ax* -f- bx n ~ x + ex"'* -j- 

 dx n ' s + &c. and a.*" -1 + bx*-* + cx n ~ s + d-z*" 4 + &c. 



In order to avoid fractions, multiply every term of the dividend by a 2 , the square 

 of the co-efficient of the first term of the divisor, and after finding 2 terms in the 

 quotient, the remainder is 



(?) = (CA 2 — Z>BA + GB 2 — OCA) ff"~* + (tfA 2 — Z>CA + GBC — ODA) X*-* -\~ &C. 



Let (ca — Z>b) a + (b 2 — ca) a = «, 

 (dA — be) a + (bc — da) a = (3, 

 (eA — £d) a -f- (bd — ea) a = y, &c. 

 and the first remainder (p) is a.x"" % + |3ar"~ 3 + yx"~* + &c. proceed with this as a 

 new divisor, and the next remainder (a) will be [(ca — Bj3) a + ((3 2 — ay) a] 



a?"- 3 + [(Da — By) a -f (j3y — ai) a] X n ^ + &C. 



Respecting this operation we may observe: 1. That were not every term of the 

 first dividend multiplied by a 2 , that quantity would be introduced by reducing the 

 terms of the remainder (p) to a common denominator. 2. When p = O, Aa:"" 1 + 

 ■Bx n ~ i -f cx n ~ 3 + &c. is a divisor of a 2 (ax n + bx n ~ l + car" -4 + &c.) ; and therefore 

 it is a divisor of ax n + bx n ~ l -f- cx n ~* + &c. unless it be a divisor of a 2 , which is 

 impossible ; consequently no alteration is, in this case, made in the conclusion, by 

 the introduction of a 2 . 



3. When p does not vanish, then every divisor of p is a divisor of ax"" 1 -f- Bx n ~* -f- 

 cx n ~ s -f- &c. and of a 2 (ax* + bx n ~ l -f- cr" -8 + &c.) ; and therefore of ax" -\- baf~ l 

 + cr" -9 -f- &c. unless a 2 = O, in which case the remainder, p, becomes as (Ba" -9 -j- 

 cx"~ s -f- &c), every divisor of which is a divisor of B# n ~* + car""" 3 + &c. whether 

 it be a divisor of ax" -f- bx"~ l -j- cx"~* + &c. or not. That is, there are 2 values 

 of the indeterminate quantity a, which, if retained, will produce erroneous con- 

 clusions. 



4. a 2 enters every term of the 2d remainder (a), and the 2 values, before intro- 

 duced, may therefore be again rejected. The co-efficient of the first term of this 

 remainder is (ca — bj3) a -f ((3 2 — ay) . a ; and, by substituting for a, £ and y, 



