INDETERMINATE PROBLEMS. 203 



of the continued frablion denoting "/{m?- — d), or </a. Thus, if 

 j2* 2 +i zz ^ 2 ; then / 1 2 —^(16 — 4)^4 — £_ and 



2 r 



~&c. 

 x zz 2, and y zz 4.2 — 1 zz 7; for 1 2.4+1 t: 49 r (y) 2 . 



It is to be remarked, that, when d zz 1, the values of x afld 

 j may be difcovered from any given number of terms of the 

 continued fraction. 



Thus, if 3^+1 ~ y 9 ', then y^ zz V{\ — 1) zz 2 — _r 



4 — i_ 



4&C. ; 



whence x zz 4, 15, 56, 209 &c. and y zz 7, 26, 97, 362 &c. 



If # zz ?n 1 -{-d, then # zz — n t and j/ — — mn — 1 3 but the 

 expreffion ax 2 -\-i zz j/*, will not be altered by changing the figns 

 of a: and j ; whence # zz n, and j/ zz mn-\-\ ; confequently, 



# and y will be determined from the continued fraction 

 m-\- 1 denoting v /(/8 a +^)» Thus, 2o# 2 +i zz j/* j 



» + _i_ 



T + &c. 



then -/20 zz ^(16+4) zz 4+j. and x zz 2, andj/zz 



a" &c. 

 4.2+1 zz 9 ; for 20.4+1 zz 81 zz (9)*. 



We may obferve, that if d zz 1 , the values of x and y, in 

 the expreffion (ffz 2 +i)x* + 1 zz j/ 2 , may be found by taking 

 an even or odd number of terms, according as the fign + or 

 — is to be adopted. 



Cor. 3. Let czo, then ax' l -\-bx zz y % ; and, in this cafe, 



* = ^=7' and .y = ~rr a > Thus » 7* 2 +4* = J^i lf m — 3» 



then x — ~~f == 2, and;/ zz -^y- zz 6. For 7-(2) 2 +4.2 zz 



c c 2 - Case 



