INDETERMINATE PROBLEMS. 203 



of the continued jraElion denoting /(?«* — ^), or ^a. Thus, if 



i2A^+i r: j' ; then /12 =v^(i6 — 4) = 4 — \_ and 



2 — I 



~&c. 



X zz ^, and / rr 4.2 — i = 7 ; for 12.4+1 ±: 49 =: (7)^ 



It is to be remarked, that, when ^ = i, the values of x aAd 



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



continued fradlion. 



Thus, if 3^'+i = j" 5" then V^ = v^(4 — i) =: 2 — ^2 



4 — I 



4 &C. J 



whence ;»: = 4, 15, 56, 209 8cc. and j/ = 7, 26, 97, 362 &c. 



If a =. 7n'^-\-d, then .v = — ;/, and y = — tnri — i ; but the 



expreflion ax^-j-i := j'% will not be altered by changing the figns 



of X and / ; whente x zz n, and y zz. mn-\-\ ; confequently, 



X and y will be determined from the continued fra(5lion 



m-\- I denoting ^{in'^-\-d\ Thus, 2oa?'+i iz j' ; 



~r+ &c. 



then v'20 =: v'(i6+4) = 4*|-i. and x = 2,andj'=: 



4.2+1 1= 9; for 20.4+1 =: 81 = (9)^ 



We may obferve, that if ^/ =: i , the values of x and j/, in 

 the expreflion (;«"+i)x» ±1 — y'', may be found by taking 

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

 — is to be adopted. 



Cor. 3. Let c =: o, then ax''-\-bx = y'' ; and, in this cafe, 



— , and j/ = ^7^. Thus, 7Jc'+4x = j' J if « = 3» 



ar rz 



then x =-—■ = 2, and jK = -^ = 6. For 7.(2)*+4.2 = 

 36 = {6y. 



c c 2 



Case 



