J.v. I. — Required ihe values of /> and q in the equation 



/'- 197'= - I- 

 By the above ruk-, 



INDETERMINATE ANALYSIS, 



Ex. 4. — Find the vahies of p and q in the equation 

 />= - '47'= '• 



Fii-ft, 



19 + o _ 



4. 1 



3 • - - 4 = 2 

 5 . i - 2 = J 



2-3-3 = 3 

 33-2=4 



'9-3 _ , 

 2 —J 



^7^" = 3 



'4 4- o 



= 3 



L4 4-3 



'4 — 1 

 = 3; -^-~- = 5 



>4 



5 • • - 3 = 2 i 



2.2 — 2 = 2; 

 5-1-2 = 3: 



= 2 

 = 5 



Whence we have, 

 quotients 



fraclion; 



n ^ 



Li 11 



3 4 



1 required. 



Having thus arrived at the quotient 8 = 2 <7, we liavu which gives 15- — 14 . 4- 



ily to connpute the fraftlons by the propolltiou above It will be obferved, that in the turcgomg examples 

 quoted ; thus, ^ have obtained for each only one folutiow ; whereas they 



all admit of an indefinite number of folutions. Moreover, it 

 does not appear, from the hrft three examples, how we 

 fhould have found the values of ^ and q, if they iiad been 

 put equal to -f- i inllead of — i . This (hail, therefore, be 

 confidered in the following propoiition. 



quotients 4 



fractions 





4S 61 170 

 II 14 39 



which laft fraction gives the values of / and q ; that 



170, ; 



= 39 ; fo 



170" — 19 • 39'' = — l> as required. 

 Ek. 2. — Find the values of p and q in the equation 



/- - 13 j^ = - I. 

 Firft, 



'- =3 I 3 • I - o = 3 -, -^ 



4- I - , 

 3 • I — 



Prop. IX. 

 To find the general values of x and y in the equation 

 .v" — N_y' = + I. from one known cafe/i — N 9" = +1. 

 In the firft place it may be demonllrated, but our limits will- 

 not allow of it, that the equation .v- - N y — I is always 

 poflible for every value of N, providing it be not a compk-te 

 fquare ; and the values of .v and y are always deducible, 

 both from ^' — N j' = i, and from /' — N y" = — I ; 

 hut if the operation above given does not produce the equa- 

 tion />' — N y' = — I, the equation a-" — N _y" = — I 

 is always impoffible. The prei'ent problem, therefore, di- 

 vides itfelf into three cafes ; wz. to find the general values 

 of X and y in the equation r' — N f, under the following 

 conditions : 



I ft, ,v- — N j^ = I , from one known cafe ^' — N 17' = i 

 2d, J,'"" — N ^y"^ = I, p'- — \\q- =—\ 



jd, x=. 

 Caf: 



N j' = I, and j;^ — N J'' = I into 



the factors J C/ 4- 7 ^/ N) (/> - 7 ^/ N) = i 

 1 (.V + V V N; (.V - y ^/ N) = I 

 then we have alfo, 



(/•4-7^/N)"■ (/.- q ^^T =■ l; 



equating thcfe, with the faftors in x and y, we have 

 (/. 4-^ V N)"' ■= X + y VN 

 ip - q ^/N)" =x - y ,/N 



, (p 4 g^/N)" + i p-q V N)'\ 

 whence .< = - — * ■ ^ ; 



2 



^iII_±-5 = s / ^'"^ ■*■ " 2VN " 



• which values of x and y will always be integral, and will be 



Here we hare, at the firll ftep, arrived at the quotient the general values fought ; and thcfe are evidently indefinite 



4. . , f . r in number, bccaufe the power m is indefinite. 



a, whence the firft fraaion, ^. is the one fought ; ior ^^,^ 2._The (ime method may he followed here, as in 



r= — I, as requireo 



the preceding culV, except that the powers inufl. be 

 G 7 



