VOL. LXXXIV,] PHILOSOPHICAL TRANSACTIONS. 415 



say nothing of other methods, may easily be investigated by the rule given in 

 page 64 of the third edition of Emerson's Fluxions ; or its equality with the 

 former series may be proved by algebra. 



On account of the sign — before x", in the last series, it may be proper to re- 

 mark, that its convergency by a geometrical progression, will not cease till 



— — -; becomes = 1, or x becomes = y'-i- ; and that when a? is a small quantity, 

 and n a large number, this series will converge almost as swiftly as the former. 

 For instance, if x be ='/4-, and n = 8, which are the values in the following 

 case, the former series will converge by the quantity x" = (v'-f)^ = -5-'-) and 



this series by the quantity = ^^ ^ := -^ ; where the difference in con- 



vergency will be but little, and the divisions by 80 easier than those by 81. 



With respect to the indices m and n, as they are here supposed to be affirma- 

 tive whole numbers, and will be so in the use about to be made of them, the 

 reader need not be detained with any observations on the cases in which these 

 fluents will fail, when the indices have contrary signs. 



It may be proper further to remark, that by putting "rz^-^ = z, and calling 

 the 1st, 2d, 3d, &c. terms of the series 



"1" mim 4- «■> . Cm 4- 9n1 . (1 — .r"^J "'" ^^' •*' ^' ^' 



m {I — .t") m (771 + 7i) . (l — .r")^ m (?« + n) . (m + 2n) . (1 — x") 



&c. respectively, the series will be expressed in the concise and elegant notation of 

 Sir Isaac Newton; viz. — ;^ ; —- — — 4- &c. which is 



»j (1 — X") 77i + n ' m + 2n 7n + 3n ' 



well adapted to arithmetical calculation. 



To come now to the transformation proposed, which will appear very easy, as 

 soon as the common series, expressing the length of an arch in terms of its tan- 

 gent, is properly arranged. If the radius of a circle be 1, and the tangent of an 

 arch of it be called t, it is well known that the length of that arch will he = t — 

 ^t^ _(_ ^i* _ ±t'' -f -i-i' — -rV'" + &c. Now, if the affirmative terms of this 

 series be written in one line, and the negative ones in another, the arch will be 



-it'- A.i' - y^^" - ^i- - -^e^ - &c. 



And if again the 1st, 3d, 5th, &c. term of each of these series be written in 

 one line, and the 2d, 4th, 6th, &o. in another, the same arch will be ex- 

 pressed thus : 



-{- 



4 



r t -\- it^ + ^i" + ^«" + -^t'' + &c. 



it' + T\t'' + tV^'' + t't^'' + -r'-rt'' -\- &C. 



•{ 



