42 HISTORY of the SOCIETr. 



Example I. To reduce the fquare root of 13 to a continued 

 fraiflion. 



Th e operation will be as under : 

 N = 13, « — 3, 2« = 6. 



P»= !,;£*= 3, R°=0. 



F= 13-9 = 4; — r-;-' + i= P' = 4,^ = i,R=2. 



F=i+/,;R' = i-h2r=3; ^-l^' = i=^ + i5 P^=3,///=r,R^=^i. 

 P"'=:4-iX(2-i)rz3; l=E' = |=i+^ P"'= 3, /."- i, R"'=2. 

 p.v_3_lX(l— 2)zr4; ^J=^=z'-=l+°-; P— 4,^-_j^R.,._Q, 



P =3-iX(2-o)=i; ^^^=^=6 + 2. pv:=:i,^'-6,R^=o. 



Here I flop, becaufe, P^ = P° = i ; and I conclude that the frac- 

 tion fought is formed by the numbers, ,«.', f/f, /*'", ;«,'% f/.^ ; that 

 is, by the numbers 1,1,1, i, 6, continually repeated in their or- 

 der. Thus, 



■* — ,1 

 i + - , 



i + _ 



6 + — &c. 

 I 



Example II. To reduce \/6i to a continued fradion: 



N — 61, n = j, in— 14. 



P° = !,/«, =7, R" =0. 



P— 61-49 = 1^; ^"=n = ^+n; F=i2,,/=T,R'=2. 

 p'^ =1 + 1x2 = 3; ■ '-T^ = T = 4 + i; P''=3,/=4,R^'=o. 



P"'=i2-4X(2-o) = 4; i^ = ^ = 3+^,; P"'=4,^"'=3.R"' = 2. 



P'' = 3 — 3XC0— 2) = 9; -^=-=1+^, P -g.^' -''^ -3- 



P' 



