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ively either odd pence, or (hilUngs : thus, 6713 pence re 

 duced, give 27/. 19X. 5^. cut off the laft, the reft are the 

 pounds required. 



To expedite the praftice, feveral compendious ways ot 

 reduftion have been invented. See Practice. 



Thus, yards are turned into ells by fubtrafting a fifth ; 

 and into ells Flemifh by adding a fifth. Ells Flen-iilh are 

 reduced into yards by fubtratling a quarter. Ells Flemifh 

 reduced to ells Englilli by multiplying by 6, and cuttmg oil 

 the right-hand figure. 



Great pounds of filk of twenty-four ounces are reduced 

 to pounds of fixteen ounces by adding one-half; and pounds 

 of fixteen ounces into pounds of twenty-four by fubtradl- 

 ing one-third. 



ReductiOxV of Decimals. See Decimals. 

 Reduction of FraSiom. See Fractions. 

 Reduction of Railoi. See Ratios, Redua'nn of. 

 Reduction of Surds. See Surds. 



Reduction of Equations. Various algebraical operations 

 are clalled under this head by different authors ; fome con- 

 fidering it to be the fame as is otherwife, and more properly, 

 ' called ihe folution of equations, or the finding of their roots : 

 fome define it to be the taking away or extermmating all 

 the unknown quantities except one, otherwife called elimina- 

 tion : others again, under this head, treat of what is more 

 ufually termed the transformation of equations ; and others 

 again apply it to the depreffing of an equation, or the reduc- 

 tion of it to another of lower dimenfions, which latter 

 feems to us the only operation that can properly be treated 

 of under the above defignation. See Resolution, and 

 Transformation. 



There are but few cafes in which the reduftion of an 

 equation can be effefted, viz. only when a known re- 

 lation has place among ft any of its routs, in which cafe 

 the equation will admit of being reduced as many degrees 

 • lower, as there are independent conditions known to have 

 place. So that if the relation be only between two roots, 

 which is one condition, the equation may be reduced two 

 degrees ; if the relation extend to three roots, it may be 

 reduced three degrees ; and fo on. 



The conditions or relations more commonly confidered, 

 are thofe in which the roots of an equation form an arith- 

 metical or geometrical progrefTion, and when an equation has 

 any numbe'r of equal roots. The two former relations feem 

 rather obieas of curiofity than utility, as it is not probable 

 tl«it an equation fliould have fuch relations obtain between 

 its roots ; but with regard to equal roots they may frequently 

 arife in the folution of various problems. When any geo- 

 metrical or phvfical problem is propofed, the number of its 

 poffibls folutions is generally limited, and therefore the 

 ultimate refult arifing out of fuch inveftigation ought to 

 be an equation, the number of whofe roots agree with the 

 limited number of folutions. But it may happen that the 

 analyft, by not purfuing the beft mode of operation, is led to 

 an equation of higher dimenfions than is requifite, in which 

 cafes, upon inveftigation, it will always be found that his 

 refulting equation has fome number of equal roots, whicii 

 being taken away, will reduce the equation to one of lower 

 dimenfions, which gives the proper number of folutions to 

 the original problem. As to the cafes in which the roots of 

 an equation form a geometrical progreflion, they occur almofl 

 exclufively in the folution of binomial equations, having 

 prime indices, a property which M. Gaufs has turned to a 

 good account in the folution of thefe equations. See Po- 

 lygon, and Reciprocal Equations. 



Nearly all other relations between the roots of equations 



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are feigned for the purpofes of framing quelliows, and ex- 

 ercifing the ingenuity of authors and their lludeiits. 



I. To afcertain whether a propofed equation has any 

 equal roots. 



Let x'" + ax"-' + /Sx"'-" 4- yo:'" ' + &;c. = O, be 

 any equation whofe roots are a, b, c, d, &c. then from the 

 known theory of equations we have 



x" + ax"-' + px"-" + yx"-' + &c. = 

 [x — a) {x ■- b) {x - c) {x — d), &c. 



And it may be fhewn alfo, that the equation m*""' -j- 

 (w- i) a-v"*-^ -f (m — 2) ^x" ' -1- &c. = 



(.1- - a) (x — b) {x — c) &c. + 

 [x - a) [x -b) [x- d) &c. + 

 {x — a) (-K— ^) \x - <') &c- T 

 (.r — b) t^x — c) [x — d) &c. ; 



that is, it is equal to the fum of all the tn equations that 

 can be formed by the different combinations of the m firlt 

 raots, taking m — i at a time. (See Waring's Meditationes 

 Algebraicae, cap. 3.) Now if we fuppoie the firll equa- 

 tion to have two equal roots, as, for example, <z = £, the 

 above products will become 



x" + a.x"-' + ^x""' + yx"-'' + &c. = 



(jf — a) (x — a) {x — c) (.V — d) &c. and 



mx"" -' + (m — l) ax"-^ -I- (ffi — 2) ISx"-' + &c. — 



(x — a) [x — a) (x - c) &c. + 

 {x —a) {x — a) [x — d) &c. + 

 \x -a) [x — c) [x — d) &c. + 

 \x — a) {x — c) [x — d) &c. ; 



where it is obvious that both the one and the other of thefe 

 equations have the fame factor, wz. (j; — a). 



If the equation had three equal roots, it is equally ob- 

 vious that both equations would have the common fa£tor 

 (x — a) ' ; and generally, if the equation had p equal roots, 

 they would both have the common faftor (x — a)'-'. 



Therefore, when it is propofed to find whether a given 

 equation have equal roots, we muft from the propofed equa- 

 tion draw the de/'ived ec\\i3X.\on as above, (which, it will be 

 obferved, is the fame as would arife from taking the fluxion 

 of the firrt, leaving out of courfe the ;f's,) and find by the 

 ufual methods, whether thefe two funftions have any com- 

 mon meafure ; which, if they have, will furnifh us with the 

 equal root fought ; and conlequently the original equation 

 may then be reduced by divifion to another of two degrees 

 lower dimenfion for two equal roots ; of three degrees 

 lower for throe, and fo on. 



Exam. I. — It is required to find the equal roots of the 

 equation x' — 484? — 128 =; O. 



Here the derived equation is 3 x' — 48 ^ o, and the 

 common meafure of thefe two funftions is x ■+ 4 ; whence 

 — 4 and — 4 are the equal roots ; the third root being 



Exam. 2. — It is required to afcertain whether the equa- 

 tion 



.V -4- 3.1:' — 14 v' — I2.V -t- 40 = O 



have equal roots, and what they are. 

 Here the derived equation is 



4.1' + 9. r' — 28 a' — 12 = 0; 



and the common divifor of the two is *• — 2 ; whence .r 

 = 2, 2, which are two of the equal roots. Divide now 

 the original equation by jr' — 4.1: -f 4, and we have 



x"- -f 7 J + 10 = o, 



whofe 



