INDETERMINATE ANALYSIS 



in order to convert — i into + i, as is obvious from but if A be not fo found, then 19 the equation impcfllble, 



infpec^ion ; and, therefore, the generll values of x and y This theorem cannot be demonarated in this place, as it 



will be belongs to the theory ot continued fraAions, and the reader 



,„ is therefore referred for a pvoof of the rule to Le Gen- 



i X = ( /» + 7 A^ N)-° 4- (/> - y ^/ N) ^^^,^ j.^^j fy^ 1^ Theorie des Nombres. 



J 2 _Ev. I.— Required the values .of .r and v, in the equation 



1 . ^ ( /■ 4- ? ./ N)- -{p-q ./ N)-" .v'_23/ = 2. 



' 2 v' ^ Here by the rule Prop. VIII. 



Cafe ^. — Here, again, we have evideatly the fame refult as /t, j_o I 23— 4' _ 



in the former cafes, except that the power m mud now be ^ — '■ = 4 i . 4 — O =• 4 ; j — T 



odd for every odd power of — i = — i ; therefore, 



V ^ (j' + gv-N) -fQ^-g.^N) 7 _ I , . I - 4 _ ^ , ^ 



V ^ ■ A^:^,; + I _ 



y - (/ + g a/N) - (/- g a/N) Having thus arrived at the denominator 2, it follows that 



L ' ' 2 ^/N (|,g equation is poflible, and the values of x and y are 



Let us now propofe an example in each of thefe three found from the fame calculation as at Prop. VIII. ; thus 



cafes. quotients 4, i, 3 



Ex. I. — In the equation p"- — 14 j' = i, we have .,45 



^ = 15 arid g = 4, to find a fecond value of p and q, or fradlions J -' —' 



of jrand y, in the equation «' — I4v^ = !• , , ,...''. 



Make whence .v :=: 5, and v = I, which gives x' — 23_y= r. 



■ , , V, , -• Ex.2 Required the values of .r and V, in the equation 



_ C5+4./h)-0;-4a-h» _^^ .,_,,,= .,. 



J Fiift 



,, = (^5-^-4^^I4r-(I5-4^-Hr ^ ^^^ 



!_ -^ 2 V14 



which give 449^ — 14 . 120' = i ; and other values may be 

 found by affuming any other power above the fecond. 



Ex. 2. — Given ^ ::= 4, and q z= j, in the equation 

 /' — 1 7 y = — I , to find the values of x and y, in the 

 equation .v — l"] y'' ^= 1. 



1.4-0 = 4; — 

 19 



Here we have. 



Haling therefore found the denominator j, the equation i» 



f _ (4 + V^l) ' + (4 ~ A^l?) ^ _ , pofilbie, and v.-e have 



I .\ _ . ^ _ 3^ quotients 4, 



i „ = (4+ ./i7)'-(4-a/i7 )'- ^ s 



L^ 2,/17 



whence 33^ — 17 . S' = i ; and other values may be found 

 by affuming any other even power inftead of the fecond. 



Ex. 3. — Given /> = 4, and q = I, in the equation 

 p~ — 17 jr" = 1, to find the values of k and j, in the equa- 

 tion x" — 17 ^'^ = — 1. 

 Affume 



r _ (4+ v'17)^^ (4- a/i7)'- 



268 



1 (4_tA/'7r-(4-./i7)'^6 



l^ 2v,/I7 



whence, 268' — 17 • 67' = — I, and other values may be 

 ebtained, by affuming other odd powers indead of the 

 third. 



Pkop. X. • whence it follows, that fince only i enters into the deno- 



minators of thefe quotients^ no ene of the propofcd equa- 

 'I'o afcertain the pofRbility or impoffibility of every equa'- tions are poUible. 



T:^.^::^^ll£rJ^i^ '"'' '° ""' ^ '^"' ' ^"'--^y — ±^^ propontion. w.may demonflrate 



,n the former cale, A ucmg < in. generally the impcrfribiUty of all ei 



The rule for this purpoie, is to conTert ^'N into a feries ^ r /f f,,nowin^ forms • 



of quotients, as in Prop. Vlli- ; ana ii A be found in the ' ° 



-denomtnator of any of the quotients, that is, if A be found x^ — {a' + 1) y' — + A 



amongft. any of the numbers, which in the propofuion x'- — {a— 1) y' = -r A 



above quoted arc reprcfentcd by n, n', »", &c. tlie cqua- •»•' —("'' + '^} y' — + A 



.tlon is poflible, and the converging fradtion correfponding x' — (a' — a) y' = + A 



to the quotient preceding this will give the values of .»■ and j;; in which A > i, and < «. 



quations falling undbr 



