VOL. LXXXVIII.] PHILOSOPHICAL TRANSACTIONS. 343 



their values, and retaining only those terms in which a is not found, and those in 

 which it is only of one dimension, we have 



ex = — £bca -f- acB 2 

 — ac 2 A 

 — Bj3 = -f- #BCA — acB 2 

 + OBDA 

 Cx — Bj3 = — «C 2 A + OBDA 

 (Ca — Bj3) a = — a 2 B 2 C 2 A + a 2 B 3 DA 

 (|3 2 — ay).A = a 2 B 2 C 2 A — 2 B 3 DA; 



therefore, those parts of (cx — b|3) * + (P Q — ay) . a, in which a is of one dimen- 

 sion, and in which it is not found, vanish. In the same manner it appears, that 

 a 2 enters every other term of the remainder <a. 



5. If the remainder o. = O, then, by the 2d observation, the introduction of o? 

 in the division, produces no error in the conclusion; and if a do not vanish, a 2 will 

 be found in every term of the 3d remainder, and may there be rejected; and v so on. 

 Thus we obtain the conclusion, without any unnecessary 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 2d division. If the difference of the highest indices of x be m, 

 the terms of the dividend must be multiplied by a* + *, the first quotient being car- 

 ried tom+1 terms. This quantity A m + 1 will enter every factor in each term of 

 the 2d remainder. 7. If it be necessary to continue the division, let 



(Ca — Bj3) . a. -f- ((3 2 — xy) . A = OTA 2 , 

 (Da — By) . x -J- ((3y — x§) . A = WA 2 , 

 (Ex — B<T) . a + Q9| — xi) . A = jbA 2 , 

 &C. 



and the 3d remainder is [(ym — (3n) . m 4. (w 2 — mp) . x~] x n ~* + \_($m — (3/)) m + 

 (np — mq) a] x n ~ i -f- [(*m — (3g-) m + (nq — mr) x] x n ~ 6 -f- &c. every term of 

 which is divisible by a 2 . The law of continuation is manifest. 



Prop. 2. Two roots of an equation of 2m dimensions may be found by the. solu- 

 tion of an equation of m . (2m — l) dimensions. 



Let x im + px™- 1 + qoo"^ -f- rx %n ^ -f &c. = O; and, if possible, let v and z be 

 so assumed that v + z, and — v + z> may be 2 roots of this equation; then, 

 as* = u 2m ± 2mzu 9W - 1 + 2m . 2ot ~ X . z 2 *; 7 *- 2 + &c. 1 



^srn-i _ ± p^«-i _L. ( 2 »> — 1) . pZV* m ~ 2 ± &C. I _- . 



9 af*»-* = ^?; 3n - 8 ± &c ( 



rx im - s = + &C.J 



2m— 1 



conseq. v 2m + 2m . ^— - . z 2 1 + z 2 "* 



+ (2m - 1) . pz \ v^ + &c. * &£? 



T ? J + r2 .2m-3 



-f &c. 



= 0; also 



