VOL. LXXVI.] PHILOSOPHICAL TRANSACTIONS. 67 



viz. if one root he x =z a; then let it contain {x — a)", where m is not a whole 

 number ; find the roots of the equations resulting from making every irrational 

 function of x contained in the given equation = O, there being no irrational 

 function of ^ contained in it ; then, if a:* be greater than the least root not = O 

 of the above-mentioned equations, the series will not converge ; but if it be a 

 series descending according to the dimensions of x, and x be less than the 

 g eatest root of the above-mentioned equations, the series will not converge. 



In interpolating seriesee to investigate the fluent contained between two values 

 a and b of the fluxion (a^t* -j- bx''+'' -{- &c.) Lv, it is preferable to make the ab- 

 scissa? begin from every one of the above-mentioned roots contained between 

 a and b. Most commonly these serieses will not converge unless x be less, &c. 

 than other quantities not investigated by this rule. 



Sir Isaac Newton gave an elegant example of this rule in the reversion of the 

 series, y = ax -[• bx'^ -{- cx^ -\- &c. from which the investigation of the law of 

 the series has never been attempted. In the year 1757 1 sent the first edition of 

 my Meditationes Algebraic'ae to the r. s., and published it in 1760, and after- 

 wards in 1762, with another part added, on the properties of curves, under the 

 title of Miscellanea Analytica, in all which was given the law of a series for 

 finding the sum of the powers of the roots of an equation from its co-efficients. 

 That great mathematician M. Le Grange and myself printed about the same 

 time an observation known to me at the time that I printed the above-men- 

 tioned book, that the law of its powers and roots, if it proceeds in infinitum, is 

 the same ; from which series of mine, with great sagacity, M. Le Grange found 

 the law which Sir Isaac Newton's reversion of series observes. In the Medita- 

 tiones the law is given, and the series is made to proceed according to the 

 dimensions of x, &c. 



It is asserted in the Meditationes, that in most equations of high dimensions, 

 unless purposedly constituted, the sum of the terms which, from the given by 

 pothesis, become the greatest, being supposed = O, only an approximate to the 

 value ax" of y in the resulting equation can by the common algebra be deduced. 

 In this case the approximate to the quantity a is to be found so near as the ap- 

 proximate value of the quantity sought requires ; or perhaps it is preferable to 

 correct in every operation the approximate values of the quantities a, b, c, &c» 

 in the series required aV -f- b'x''+'' -\- cx''+^'' -\- &c. In the equation the quantity 

 z + e may be substituted for Xj and from the equation resulting a series express- 

 ing the value of 7/ may be found, proceeding either according to the dimensions 

 of the quantity z, or its reciprocals, according to the conditions of the problem. 



If there be more than one (rz) equations having n -J- 1 unknown quantities, 

 X, y, z, &c., in each of the equations, for y, z, &c. write respectively at", 

 Pl'ocT, &c. ; and suppose the terms of each of the equations, which result the 



K 2 



