R A T 



RAT 



RATIONABILIBUS Divims, is a writ that lies where 

 two lords have figiiorics joining together, for him that finds 

 his wafto encroached upon, within the memory of man, 

 againft the encroacher, thereby to reftify the bounds of the 

 fignories : in which refpcft, Fitzherbert fays, it is of the 

 nature of a writ of right. 



RATIONABILIS Dos, a third part of fuch lands 

 and tenements as the hufband was feifed of at the time of 

 the efpoufals, with which his wife was formerly endowed by 

 the common law, if no fpccific dotation was made at the 

 church porch. See Dowkr and Jointlke. 



RATIONAL, Reasonable. See Reason. 



Rational Fable. See Fable. 



Rational Frailions, in Arithmetic and Anal^u, are thofe 

 fraftions into which no furd or radical quantity enters ; as 

 + 1 x' + -v- + ax 



whence r and s may be determined, and we (liall thus obtain 



7 19 



&c. 



8' 27' k' + i' x' + *x' + ex" 



The dccompofition of rational fraftions into fimple frac- 

 tions, that is, the dccompofition of them into other frac- 

 tions whofe fum is equal to that propofed, is an important 

 problem, as connected with the integral calculus, or inverfe 

 method of fluxions, which was tirft inveftigated by Leibnitz, 

 but has fince been much extended and fimphfied by the re- 

 fearches of Euler, La Grange, La Croix, and other eminent 

 analyfts. 



The dccompofition of numeral fraftions into their par- 

 tial fradlioiis, is, perhaps, rather a fubjeft of curiofity than 

 utihty, yet as conneAed with, and leading to, the dccom- 

 pofition of rational algebraic fraftions, it may not be amifs 

 to give here a fketch of the procefs by which it is accom- 

 plifhed, previous to entering upon the latter fubjeft. 



On the decompofttion of rational numeral fradions, into others 

 having prime denominators. , 



It is to be obferved, that this can only be effefted in the 

 cafe of a compofite denominator, or rather, there will be no 

 difficulty in any other cafe, as it will require only a fepara- 

 tion of the numerator into any parts at pleafure, the fum ot 

 which is obvioufly equal to the fraftion propofed ; we fliall 

 therefore only confider thofe frattions having compofite de- 

 nominators, which are to be refolved into others having prime 

 denominators ; and even in this cafe there may be fraftions that 

 will not admit of decompofition, as will appear from what 

 follows. This decompofition is effefled, when poffible, by 



means of the indeterminate analyfis ; viz. let - be the given 



n 



fraftion, and fuppofe, in the firft inflance, tliat its denomi- 

 nator confifts of two prime factors, or ti — ab, it will then 



be to find 



ab 



= ^ + -1-, or a q + bp 

 a t> 



I ; p and q 



being the required numerators of the two partial fractions, 

 and which values of jJ and q are eafily found from the above 

 equation a q -\- b p = m, on the principL-s explained under 

 the article Indeterminate Analyfts, fubjeft however to 

 the fame limitation there mentioned ; i-ix. the above equa- 

 tion is always poflible, provided m > a b — a — b, but in 

 other cafes it may or may not admit of folution. 



If the given fraftion be - - — , then we may firfl 

 ° a b c 



refolve it into two fractions, and one of thefe into 



two others ; thus, let -'"- — ^.- + ^ , then we have 

 a b c ab c 



ahq + cp— m, from which equation/) and q may be found. 



Again, let ^ = 1- -r-, which gives a s ■{■ r b =■ p, 



° aba 



a b c a 



s n 



-r- + --, as required. In all thefe cafe 



it is obvious, that the frat'tion may be deconipofcd into par 

 tial fractions, in as many different ways as the indett-rminat^ 

 equation on which it depends admits of different anfwers. 



Example — Find two fraftions, having prime denomi- 

 nators, whofe fum (liall be equal to , or to — . 



35 7-5 



Let the required fradlions be — -)--- , then — -- 



7 5 35 



I, and 



— ; therefore 5/1 4- 7 y = 19, whence p 



q ~ 2 (fee Indeterminate Analyfis) ; and confequently 



I 2 



the required partial fraftions are — and — . 



7 5 



Example 2. — Find three fraftions, whofe fum is equal to 

 401 



315' 



The three faftors of 315 are 5, 7, 9, which are of necef- 

 fity the denominators of the required fraftions. Suppofe 



then firft, that = -i~ -\- J-, whence we have q S 4- 



315 IS 9 -^ 



35 y = 401, which gives p — 29, and y = 4; therefore 



401 29 4 A ■ , 29 r 

 -- — =--4--r. Agam, let -^ = — 

 3'5 35 9 35 7 



+ 



Sr + 



"J s = 29 ; this gives r = 3 and s 



401 



2 3 



= — + -- + 

 5 7 



= 2, fothat-^= ^- -I , 



35 7 5 



-, as required. And 



and confequently 



315 5 7 9 

 in the fame manner the decompofition may be obtained in 

 any other cafe, which falls within the limits above ftated. 



On the decompofition of rational algebraic fradicns, and its 

 application to the integral calculus. 



N « 4 ix -i- fx" -f </x' 4- &c. ../x"-' 



D "~ a' 4- b'x 4- c' x' 4- d' x^ 4- . . . .p'x"^' + q'lp" 

 be any rational fraftion, whole decompofition into fimple 

 fraftions is required, and whofe numerator is at leatt one 

 degree lower than its denominator, to which form it may 

 always be reduced by divifion, if it (hould prefent itfelf 

 under a different form. 



The denominator ot this fra6tion, from the kno-.vn theory 

 of equations, may be fuppofed to be made up of as many fim- 

 ple faCiors as is equal to the highefl power of j: contained 

 in it, and which fatlors may be found by determining the 

 roots of the equation formed by putting the whole deno- 

 minator equal to zero. Let therefore a' -j- b' x -f c' x^ 

 -i- d' X -\- , . . . x"" = o, and fuppofe the roots of this 

 equation to be r, r', r", r"', r'" , &c., and we fhall have a' + 

 L' X + c'x' + d<x' + x"" = (x - r) { X - r') (x - r") 



{■ 



_ r'" 



), &c. 



'N 



And therefore our propofed fraftion y: ""-ay now be put 



under the form 



N a + bx -\-cs'^ + dx'' + 



&c. 



D " (x - r) (x _ r*) (x - r") (x - r"')' 



and thefe faftors are now to form the denominators of 



the 



