66 PHILOSOPHICAL TRANSACTIONS. [aNNO 1786. 



or = e, when z is very small ; and then write z + ^ fo** -^ »" the quantit) , and 

 reduce it in the former case into a series proceeding according to the dimensions 

 of e, in the latter case according to the dimensions of z, and there will arise '2 

 serieses, of which the fluents properly corrected, viz. by adding the fluent con- 

 tained between the values a and e to the latter, and that between a and z to the 

 former, will give the same fluent. The first term of the series, in which e is 

 supposed very small, will be the fluent of the given fluxion, when x = z. 



If a fluxion p.r, where p is a function of x, be transformed into another ai, 

 where q is a function of z, and they be reduced into serieses a and b, proceed- 

 ing according to the dimensions of x and z respectively ; find a. and tt, corres- 

 pondent values of the quantities x and z ; then in ascending serieses, if a has a 

 less ratio to the least root of the equation p = O, than tt has to the least root of 

 the equation a = O, the series a (exceptis excipiendis) will converge swifter than 

 the series b. 



Dr. Barrow, in some equations, expressing the relation between the absciss x 

 and ordinate 2/, found 7/ in the first 2 terms of x, viz. 3/ = a + ^^) which is an 

 equation to the asymptotes of the curves. Sir Isaac Newton, from an algebrai- 

 cal equation given, expressing the relation between t/ and x, found a series pro- 

 ceeding according to the dimensions of x, expressing 1/ in terms of x. M. 

 Leibnitz performed the same problem by assuming a series ax" -j- bx" + '' 

 -f- ca;"+*'' -|- &c. with general co-efficients, and substituting this series for 3/ in 

 the given equation, &c. from equating the correspondent terms he deduced the 

 indexes and co-efficients. M. De Moivre, Mr. Maclaurin, &c. observed, that 

 when the highest terms of the given equations have 2 or more {m) divisors equal; 

 for example, (y — ax")'" ; to which we must add, and when a value of 1/ in this 



r 



case is required nearly equal to ax", a series ax" -\- bx "* -f- &c. is to be as- 

 sumed, whose indexes differ only by -, &c. if otherwise they would differ by r. 

 This reduction seldom answers any other purpose than finding the fluents of 

 fluxions, as / z/.r, &c. ; or the asymptotes, &c. of curves, which depend on 



some of the first terms of the series ; but will very seldom be used for finding 

 the roots of an equation. The rule of false, or method given by Vieta, will 

 ever be substituted in its stead. 



The values of x may be required between which the above-mentioned series 

 Aoif + Bx"+'' -|- ca?"+^'' -j- &c. will converge, as the infinite series answers no pur- 

 pose when it diverges. First, if an ascending be required, write for y in the 

 given algebraical equation an infinite quantity, and find the roots of x in the 

 equation thence resulting p = ; which for y write in the same equation, and 

 find the roots of x in the resulting equation which contain irrational quantities, 



