432 PHILOSOPHICAL TRANSACTIONS. [aNNO 1753. 



other things remaining as in prop. 1 ; then shall the fraction 



— ^t_fff formed as there described, be such, that - — -r^-:.r^^^ = a. 

 ibc ' C^bcj^ 



For because bb -^ I = ace, then ace — bb = 1; and squaring, 

 b* — 2aiV + aV= 1. 



Hence, as in the foregoing, it will follow, that 



ebb + accj^ — 1 



= a. 



C^bcJ" 



Prop. 3. Let the fraction - be such, that — ~ — = a, i. e. bb = ace + 1; 

 also let - be another fraction, having the same property with -, i. e. such, that 

 dd = aee + 1. Then, if from the fraction -, and the two others mentioned 



be 



in prop. 1, viz.—, and 7, a new fraction be formed, in the same manner as the 



fraction — -7 — was formed from -, and the same two - and 7, which fraction will 

 2bc c ac b 



be , , ; this new fraction shall have the sanie property with the other two 



b J d . (bd + ace)'' — 1 



-and -,i.e.-^^-5-^_-. = a. 



Prop. 4. The same things being supposed as in prop. 3, except that bb, in- 

 stead of being equal to ace + 1 , as there, is equal to ace — 1 , or bb -{- 1 ^ ace; 



it will follow, by the like steps as in prop. 3, that — - "^ = a. 



Prop. 5. If likewise cP be equal to aee — 1, as well as b^ = ace — 1, all other 

 things remaining as in prop. 3, then shall {bd -\- acey = a X {cd -{- bey + 1» 



Cbd + ace)* — 1 _ 



Prop. 6. But if i^ = ace + 1, and d^ =■ aee — 1, all other things remaining as 



(bd + acej'^ + 1 

 'bTJ' 



Now, let a be any number proposed, and let the fraction - be such, that 

 either — '^^— = a, or — — = a, and take the fractions — and-r, before described; 



cc cc ac o 



then the series of fractions converging to \^ a, will be as follows: 



-, —\- = the first term of the series. 

 b' aci c 



*i + ^ = ^the2dterm. 1 



, 2oc e Every term is formed from the preceding; and 



bd + ace L tV, <kA t b c 



Id ^rt7 ~ g term. ^ ^j^^ ^^^ fractions — and - , in the same manner as 



J-2L^ = - the 4th term. the second from the first, and these fractions. 



cf+bg k 



&c. in infinitum, j 

 And from the foregoing propositions it follows, 



1. That if — "Ili = a, then every fraction of the series shall be such, that if 



in prop. 3. Then shall {bd + acey + \ = a X (cd + bey, i.e. '^^-^-' = a. 



