LIMITS. 



\)e the limits of the value of x, that is, a value of .r will 

 always be found between thefe two quantities. 



Let .r- - A ^i"- + B .V—' - C x-" -•' + &c. + N = c. ; 

 and fuppofe that, by fubflituting any quantity p, inttcad of 

 X, we have 



p-" - Ap" ' + B/)-" - - c/-" ' + s:c- -T >-' = R ; 



and by fubftituting another quantity, q, for it, we obtain 



5- -A 9"- + Bj-^-C^--^ + &c. N = -S: 

 then, t fay, that there is at leaft one real value o£ x between 

 the limits /. and 7 ; that ij, x is lefs than the former, and 

 greater tliau the latter. The truth of llic propofition, how- 

 ever, is better demonftraied from a partial than from a 

 general example. 



Let us, therefore, afiumc the equation 



.t^— ij.v' + 7.V — 1 =0; 

 here, if we fubftitute .v = 2, and x = 20, wc have a 

 refult in the firll cafe = - 3 1 ; and in the fccond = -r 

 -939 ; and it remains to be (hewn that there is, at leatt, one 

 real value of. x coniprifed between thefe limits. For this 

 purpole, the equation may be written 



..'+ 7^- - ('3' + 



found to be negative when x = 2 ; 



the refult is pofitive, and x c= o, making it negative, a real 

 value of jr mull he between the limits/; and o. Again, the 

 above equation may be converted into anotlier, having th<? 

 fame roots, only with contrary ilgns, by writing — y for 

 X. And let us fuppofe, in the full place, that ni is even^ 

 then the transformed equation will be 



J- + Ay'"- + B.r' ' + Cj"-' &c. + N = o ; 



and, conftquently, N will ilill have, with regard to^"*, the 

 fame fign, which, as above, we fuppofe to be negative ; then, 

 if y be fuch as will give a pofitive refult, and .i- =: o giving 

 a negative, it follows, that a real value of y will be found 

 between the limits g and o ; and, conlequently, in the 

 equation propofed, a real root is comprifed between the 

 limits — g and o. 



But if tlie power m be odd, then the transformed equation 

 will be 



+ Ay- 



- C>"-5 -f &c. + 

 + Cy-' ± &c. + 



N 



N: 



: O, 



which quantity is 



'I'hat is, in the lirlt cafe, we 



but pofitive when x = ; 

 liave 



{x-> + 7-v) < ('i-^-' + 0' 

 and in the latter 



(*■' + 7-^-) > (I.?-*"' + ')• 

 Now, it is obvious, that each branch of thefe cxpreffions 

 will increafe as .v is augmented, and that they will likewife 

 be each diminifhed as .v is diminiflied. Let us, theretorc, 

 conceive x, in the firft cafe, to be fucceffively mcreafed by 

 any fmall quantity, till it arrives at the value of .r in the 

 fccond cafe. Then, fince .v' + T x, which was at iirit lets 

 than H I- + 1, is now become greater than 13 x' + i. *} 

 muft neceffarily have palfed through that Hate, in whicli it 

 was neither greater nor Icfs ; that is, the two branches mutt 

 have palfed through that Hate in which they were equal ; 

 but when 



x' + 7.r = (13X' + I), 



we have alfo 



x'^ — IS X- + •; X — 1 = o ; 



and, confequently, this value of x is a real root of the cqua- 



lion propofed. , , , • • 1 



This reafoning, though employed only in a particular 

 cafe, is equally applicable to our general equation : for, by 

 puttinT the pofitive part of the equation = P, and the 

 negative = Q ; alfo, fuppofmg /> to be that value of ..• 

 which renders the refult negative, or, which is the fame, 

 v*-hich gives P < Q ; and y that value which makes 

 Y •-, Q, then we mav conclude the fame as above, that P, 

 from being lefs than Q, having paifed to that ftate in wliich 

 it is greater than (>, there mull neceffarily be a real value of x, 

 between p and 5,"which renders P = Q ; or the propofed 

 equation = o. We may alfo afcertain the limits of x be- 

 tween o, and fome real quantity, pofitive or negative. For 

 example, in the general equation 



X-" - A*"— + B .v"-^ - C .x"-' + &c. + N = o, 



T Bj'-" 



± D/ 



and, confequently, y" and ^ N, have not the fame fign 

 with regard to each other. If, therefore, now, any value y 

 can be found, fuch that the refult may be negative, a root 

 of this equation will be found between the limits g and o, 

 and, therefore, in the original one between — j and o. 



2. The greateil pofitive root of an equatioi. is always 

 contained between the limits S (- 1 and o ; S being the 

 greateil negative co-efRcicnt that enters into the equation. 



In order to prove this, we muft demonftrate that in any 

 expreflion 



.v" + A .r"--' + B-t""-' + Cr""-' &c. N. 



The firll term may be made to exceed the fum of all the 

 other terms. Now, it is obvious, in the firft place, that 

 the cafe which prelents the greateil difficulty, it that in 

 which all the co-efficients are made negative, and each equal 

 to the greatefl ; let, then, S be the greateil negative co- 

 efficient, it is to be demonftrated, that fuch a value of x 

 may be found as will render 



.i" > S (.x'"-' 4- .x"-' -1- .v"""' + &c. -f 1). 



Or, fince the part within the parenthelis is equal to - 

 we have to fiiew, that we may find .v fuch, that 



vS x" S 



X- J 



> 



S {x--l) 



— i , or . 



> 



X — I 



Now, this will be manifellly the cafe, if we make 

 S .v" S 



-, or X 



S + I. 



itis obvious, that by taking x = o, the refult will be nega- other negative 



It is therefore obvious, that this value, fubftituted for x in 

 the propofed equation, will give a polltive refult ; whereas, 

 X = o gives a negative refult : tlierefore, from what is 

 ihewn above, a real value of x is found between the limits 

 8 + J and o. If the foregoing equation be converted 

 into another, with the figns of the roots changed, and if R 

 in that equation be the greateil negative ce-efficient, then 

 — (R -(- l), and o, will be the limits alfo of the greattlt 

 negative root. 



it follows, immediately from what is fhewn above, that 

 every equation of even dinienfions, having its laft term nega- 

 tive, has at leaft two real roots, the one pofitive and the 



live or pofitive, according as N is affedted with the fign 

 er + . Therefore, if, in the firft place, we find p fuch tliat 



It may alfo be readily demonftrated, upon Cmilar princi- 

 ples, that every eqiKition of odd dimenfwna has at leaft one 



real 



