VOL. LXVIII.] PHILOSOPHICAL TRANSACTIONS. 383 



tainly requires some restrictions. For example, the roots of the quadratic equa- 

 tion x'^ — 2x — 3 = are 3, and — 1 ; multiply the terms of this equation 

 into the terms of the arithmetical progression I, 2, 3, respectively, and the 

 resulting equation is 1 X x^ — '1 x 2x — 3 x 3 = O, the roots of which are 

 2 + V 13, neither of which are between the roots of the given quadratic. 

 Again, suppose the roots of the cubic equation x^ — px- -\- qx — r = to be a, 

 b, — c, then it is possible that the equation (/ + 3m) X x^ — (/ + 2m) X P^^ + 

 (l X m) X qx — //• = O may have no root between the quantities b and — c; 

 and in general, if the roots of the equation (a) a," — px""- ' + qx"-* &c = 

 be supposed o, b, c, — d, — e, — f, &c. w here a i s the greatest root, b the 

 next, and so on in order, the equation (b) (/ + nm) X x" — [/ -|- (n — 1 ) 

 m] px"~' + [/ + {n — 2)7n] qx''~'^ &c. = O will not necessarily have any of 

 its roots between the roots c and — dof the original equation. 



^ 3. It will not be difficult to see the reason of this, if we examine the 

 demonstration usually given of this 2d proposition. The roots of the biqua- 

 dratic equation x* — ax^ -j- bx' — co.^ + d = O are supposed to be a, b, c, d, 

 and the results which arise by successively substituting them for x in 4x^ — 3Ajr* 

 + 2Ba: — c are supposed to be — r, -f- s, — d, + z. From which Maclaurin 

 concludes, that when a, b, c, d, are substituted for x in the quantity (/ -j- 4m) 

 X x^ — {l-\- 3m) X Ao;^ + (/ + ^m) Bx"^ — (/ + to) c:r + /d, the quantities that 

 result will become — wjrx, + m&x, — mrx, + mzx, where, says he, the signs 

 being alternately negative and positive, it follows, that a, b, c, d, must be limits 

 of the equation (/ + Am) X x^ — (/ + 3m) ax' + &c. = O. 



Here it is taken for granted, that the quantities — otrot, + msx, — mTx, 

 + mzx, are alternately negative and positive, which is not true, unless the 

 roots a, b, c, d, be either all positive or all negative. For suppose a, b, c, to be 

 positive* quantities, and da negative one; then the four results will be — mB.a, 

 -j- mib, — mTc, — mzd. 



^ 4. In general, the roots of the equation ?ix""" ' — {n — 1 ). px'- ' -j- (n _ 2), 

 qx"~' are always between the roots of the equation (a), because the roots of this 

 last equation substituted successively for x in 7ix "~ ' — (n — l). px"'" -\- &c. 

 always give the resulting quantities alternately negative and positive; but when 

 the least of the affirmative roots, and the greatest of the negative roots of the 

 equation (a) are substituted in (b), the quantities that result will necessarily have 

 the same sign, and therefore it is possible that no root of the equation (b) may lie 



* Philos. Trans, vol. 36. Mr. Maclauiin, who is here very diffuse on tliis subject, never mentions 

 any exception of this sort. In his Algebra, art. 41', part 2, he says, he shall only treat of such 

 equations as have their roots positive ; but it may be observed, that his reasoning from art. 45 to 50 

 holds in all equations, the roots of which are real. The theorem in p. 182 of that treatise is not 

 treneral, though applied in the 11th chapter to the demonstration of Newton's rule for finding 

 impossible roots in all equations. — Orig. 



