his Tort of the sociEtr. 2 i 



2. Take P' = N — « 2 for a divifor, and 2» — R° = 2n for a 

 dividend. Let the quotient be yf, and remainder R'. 



3. T a ke F = P ° — (R ° — R') pJ; for a divifor, and 2n — R' for 

 a dividend,. Let the quotient be yf and the remainder R''. 



4. Take P'" = F— (R'— R")Xy" for a divifor, and 2n — R' 

 for a dividend. Let the quotient be yJ'\ and the remain- 

 der R'". 



5. These operations may be continued without end ; 

 the divifor F F being found from the formula P^ rr 

 Pr-» ___ ^-2 _ j^-zj x ^-1 . the C orrefponding dividend 

 being 211 — RP~~ X ; and the quotient of the di virion being 

 denoted by yf, and the remainder by Rp. But it will only be 

 neceffary to continue thefe operations till we arrive at a value 

 P'z: P° zz 1, which will always neceffarily be the cafe. After 

 this, the feries of numbers, yf +I , yf* 2 , y p+3 i will neceffarily be 

 the fame as the numbers yJ, y ff , y!", &c. ; yf> continually re- 

 peated in their order. 



The rule may be fhortly expreffed in algebraic language, 

 thus : 



P° = 1; ft = flXl+R°-«; 



F = N — *z 2 ; 2«-R° = 2»=P'X^'-fR' ? 



V = 1 — ^7R ° — R') = 1 4-^'R'; 2ti — R' = Y X y" + R" ; 

 F" - F — y,\R' — R') ; 2»— R' = P'" X y!"+ R"; 



Fv = r — yl" (R' — R'") j 20 — R" = P ,v X y, lV +R 1Y - 3 . 



and fo on. 



Having thus found the numbers, y,, y\ yf, y,°, we fhall have 

 the continued fraction fought, 



**' + — 



*** + - 



/»'"+&<:. 



¥f 



And the fraction may be continued indefinitely, by repeating 

 the denominators y\ y%"> y p , continually in their order. 



Example 



