4Sa PHILOSOPHICAL TRANSACTIONS. [aNNO 1778. 



of the -th power of the binomial quantity l — x, or, according to Sir Isaac 



Newton's notation of powers, the quantity {1 — x) ", will, according to that 

 celebrated theorem, be equal to the infinite series 



1 + - AX H -^— BOT^ H — cx^ H 7 — Bx^ H — E^^ &c. ; m which 



series the capital letters a, b, c, d, e, f, g, h, &c. stand for 1 and the co-effi- 

 cients of X, a.', x^, X*, x\ x^, x\ x**, &c. Now it is evident, that the generating 



^ . m + n m + 2n m + 3't m + in m + 5n m + 6n m + 7" „ 1 • 1 



fractions — - — , — - — , — , -, — -. — , —7. — , —z — , — r — , &c. are derived 



2/4 3» 4« 5n On 7« 8» 



from -, and from each other, by the continual addition of n to both their nume- 

 rators and denominators. Therefore, though they are greater than they would 

 be if m was subtracted from the numerator of each of them, that is, than the 



- . n 2n 3n in 5n 6n In c a .1 .1 ^1 1 



fractions -, -, -, -, g;^, -, -, &c. and consequently, than the equal cor- 

 responding fractions i, \, \, x, a, &-, \, &c. ; yet the farther we go in the series, 

 the less is the proportion in which they exceed the latter fractions; insomuch 

 that, if we go far enough in the series, we may find terms in it whose proportion 

 to the corresponding terms in the series 4-, .§-, -|-, ±, ^, &-, i, &c. shall approach as 

 near as we please to a proportion of equality. And, by taking n of a very great 

 magnitude in comparison of m, we may even make the first terms of the series 



m + n m + 2n to + 3n m + in m -\- 5n m + 6n m +7" o 1 1 



-if-' -3—' -^^' '5—' -6^' -7-' "8—' ^''- approach very nearly 

 to an equality with the corresponding terms of the series J-, a, ^, a, |., _6., x, &c. 

 which are the generating fractions of the proposed series x -\- \x'^ -\- ^^ -f- -r^* 

 -|- :Lx' -\- :yxP -\- -fr' -|- -^^ + &c. In order to this, let m be taken = I , and 

 n = 1 ,000,000,000,000, that is, = a billion, or the square of a million, which, 



to avoid the frequent repetition of so many cyphers, we will call b. Then will 

 . ; ,, >_i , , , 1 ,1 + * „ , 1 + 26 3 



y\ — X) ■,c»o,ooo,ooo,ooo^ Qr (l — X) b , 00. ^ \ -\- - l^X -\- TiX' -\ — — CX 



-|- ■^— ^— Dx* -| -^ Ex^ &c. which, on account of the great magnitude of b, 



2b, 3b, 4b, 5b, 6b, 7b, &c. in comparison of 1, will be nearly equal to (though 

 somewhat greater than) 1 + r -^•^ + 26 ^^^ "^ 36 ^^^ "*" 46 °^^ "'" 56 ^^^ ^^' 

 or 1 + "I + ^ + ^^ + ^ + ^ + &c. Therefore, multiplying both sides by b, 



— 1 



we shall have b X (l — x)~^ nearly = b -{- x -{■ ^x'^ -\- ^-.i^ -|- J-x* + ix' + &c.; 

 and, subtracting b from both sides, the proposed series x + i^'^ + -j^^ + -r^'* 



-|_ xjc^ -f &c. will be nearly equal to b (-3-)' — /^- We must therefore first 

 subtract x from 1, and then divide I by the remainder, which will give us a quo- 

 tient equal to — — . And, having found this quotient, we must extract its ^th, 



