VOL. XLVIII.J PHILOSOPHICAL TRANSACTIONS. 433 



To find - such as makes bb — 1 = ace, i. e. ace -\- I = bb, recourse must 

 be had to Lord Brouncker's method in Dr. Wallis's Commercium Epistolicum. 

 from the square of its numerator be taken a unit, the remainder, divided by the 

 square of its denominator, shall be equal to a. 



For, by prop. 1, the fraction - shall be such; and by prop. 3, the next frac- 



tion- shall likewise be such; and so all the following terms. 



Example. Let a = 2 ; then the first fraction, i. e. that in the smallest numbers, 

 -, that makes ~ = 2, is when 6 = 3, and c =. 2 ; so that 



c cc 



c b t b ■) 



And the terms following the first -|-, are -j4. j-%. 4^-1-. 4 ff^. &c. 



2. But if := a, i.e. if the first fraction - of the series have the square 



cc c ^ 



of its numerator a unit less than ace, the multiple of the square of its denomi- 

 nator by the number a ; the 2d term shall have the square of its numerator a 

 unit greater than the said multiple of the square of its denominator ; and the 3d 

 term shall have the said square a unit less, and so on alternately. For, by prop. 

 2, the second term - shall be such, that — ^^^~ = a: and therefore, by prop. 4, 



the 3d term - shall be such, that^^ = a. And by prop. 5, it follows, that 



the next term r shall be such, that — -7?- = a ; and so on alternately, by prop. 



4 and 5. 



Example. Let a = 2: then the first fraction - that makes ^2, is when 



Z) = 1 , and c = 1 . So that 



^ . y I - are 4-. -1- } -{-. And the following terms are a. -f. -f-J. -fi-. -fA &c. 



But if a be 13, then the fractions will be '-/• -rV- w] V • -frl- Wrr • &c. 



3. But if the fraction - be such, that — ^^^ = a, and if the fractions -, ^ 



c ' cc ' ac' b 



be taken, from which the series is to be formed, as has been described; then if 

 the first fraction of the series be made not -, but some fraction -, such that 



c e 



— — = a; then shall every term of the series be such as the fraction -, i.e. the 

 square of the numerator being increased by a unit, and the sum divided by the 

 square of the denominator, the quotient shall be equal to a. For since bb = 

 ace -^ 1, and dd = aee — 1, by prop. 6 it follows, that the next term 



- shall be such, that — = a; and so on for every term. 



5 ^^ _ 



Example. Let a = 2, -~ -; then will — = -, and r- = -, and let - =: -; 



' 'c=2' flc4 6 3' el* 



then p - } - are -.- } -. And the other terms are -J-. \^. \^. \^^ . &c 



• VOL. X. 3 K 



