±2 HISTORY of the SOCIETY. 



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

 fraction. 



The operation will be as under : 

 N = 13, n = 3, ±n = 6. 



P'zr i, ^ = 3, R°zzo. 

 P=i3-9= 4 ; fc£±S=i+i P 2 4,^,,R' S , 



P' =-1+^ = 14-2 =33 6 -=^ = ± =l + l. F~3,^ = i,r4i. 



P'"=:4— IX (2— i) = 3J £^! = i=i+|5 F"=3,^I,R W =2. 



P- 3 -iX(i-2)r 4 ; 6 -=^-=^i-r-? ; P" = 4 ,/*' v =i,R"=o. 



P ■' = 3-ix(2-o)=ij^ = ^ = 6 + -°j P v =i,^ = 6,R v =o. 



Here I flop, becaufe, P v = P° = 1 ; and I conclude that the frac- 

 tion fought is formed by the numbers, yJ, yf, p!" t p", ys ; that 

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

 der. Thus, 



I + T + ± 



1 



1 



i + — 



6 + — &c. 



1 



Example II. To reduce v/61 to a continued fraction : 

 N = 61, « = 7, 2;/ = 14. 



P° = i,i» = 7, R° =0. 



14 — R° 14 ■ 2 tv / n / 



p' =6!— 49 = 12; - ± ^--r, = I +r 1 - p = ' 2 .f- = i,R =2. 

 p = 1 + 1x2 = 3; • I - i p = 7 = 4 + i; P" = 3.^ =4,R' =0. 

 r= I 2- 4 x(2-o) = 4 ; ^=^ = 3+|; P"' = 4.^ = 3.R"' = 2. 



p.v = 3 _ 3 X(o-2) = 9; 2<=?-| = , + |; r* = 3,pr 5fc»"* 3- 



