INDETERMINATE ANALYSIS. 



Cor. 2. — It is alfo deducible from fimilar principles, that 

 the following equations are always pofTlble ; N being a 

 prime number, of tUe form placed oppofile the refpedtive 

 equations. 

 x' — N v' = — 1 poffible when N of tlie form 4/1+1 



-'-Ny= - 2 N 8n -H3 



' ^ H y' — 2 N 8n— I 



. i in a fimilar manner, are deduced the three following 



...^orems : 



1. If M and N be both of the form 4/1+3, and not 

 equal to each other, the equation 



M x' - N _y'- = + I 

 is always poffible in integer numbers: that is, under one 

 or other of the figns + or — . 



2. If M and N be both of the form 4 /! + 1 ; then 

 one of the equations 



.r'- - M N^.= : 



M. 



N y- = + 



will always be refolvible in integers. 



3. If M and M' be two prime numbers of the form 



4/j + 3, and N a prime number of the form 4/1 + i, 



it will be always poffible to fatisfy one of the equations 



' r N .f= - MM'/= + I 



\ M A- - M' N^' = + I 



I M'.v'-- M N^'= + I 



Pkop. XI. 

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

 X - N/ = + A ' 



from one known cafe p^ — N 9' = + A. 



Find the values of m and n in the equation w^ — N /!" = I, 

 i>y Prop. VIII. ; then it is evident that the product 



(;,' - N 9 ) X (,7r - N /! ) = + A ; 

 and it will be found, upon the developement of the following 

 formulas, that 



,A-N,, X „.--N-) = {^l^r^l^ 



we have, therefore, only to make, 



+ Ny/r r. <=/>»! — N jB 



rx=//n+Ny/r^^ Cx=p» 

 \y=pn^qm lyr^pi 



and having before ihewn Iww to find the general values m 

 and n in the equation m — N «• ^ i ; it is obvious, that 

 by means of thefe formulae we may derive different values 

 of .r and y, in the propofed equation, to any extent at 

 plcafure. 



Cor. — It appears alfo from this propofition, that if the rule 

 in the foregoing one give p — N y' = — A, when the equa- 

 tion propofed be + A ; that this may be converted to the 

 latter fign by means of the equation 



m'-N/r = - I. 



Ex. I. — Given the values of p- and q, in the equation 

 p — qq^ ^z 2; -viz.p — 3, and y = 1, to find the general 

 values of X and J', in the equation x — "] y- :^ 2. 



Firil in the exprellion m — y n =■ 1, we have w = 8, 

 and n — J ; whence by the above formula is obtained 

 f .V = /, ,« + N y « = 3 . 8 + 7 . I . 3 = 3 or 45 

 ly — p n ± qm =3.3+ 1.8 =lori7 

 fo that the fecond values of x and y »re, x = 4J, and 

 y = iq ; which give 45- — 7 . 17' = 3. And afl'uming 

 thefe again as new values of^ and^, other values x and jf may 

 be found in infinitum : or the original values of ^ and q may 



be retained, and new values found for m and n, which anfwere 

 the fame purpofc. 



Ex. 2. — Find the gener.il values of .r and j', in the equa- 

 tion .v" — I i_j' = 5 ; the known cat being />' — \i q- — 5, 

 or^ r= 4, and g = i. 



In the equation m' — 11 /i'= i, we have m^= 10, and;/ = 3, 

 therefore by the formulae 



f v = /> /,; ± N 9 /i = 4 . 10 + II. I . 3 = 7 or 73 

 \y — pn± qm =4. 3+ i . 10 =2 or 22 

 that is .r =: 7, and jr r= 2, are two new values cf x and j', 

 as are alfo x :::; 73, and^ = 22 ; for each of thefe give 



in- 11 •"- = 5 



and in the fame manner, other values may be found to any 

 required extent. 



As our limits will not allow of a very full and explicit in- 

 vefti^ation of the feveral elegant rules that have been intro- 

 duced into the indeterminate analyfis by Euler, Lagrange, 

 Legendre, and other diftinguifr.ed mathematicians'; we muft. 

 refer t!(e reader for the inveftigation ef the methods em- 

 ployed in the following propofitions to the authors above- 

 mentioned, and muft content ourfelves with barely Rating 

 the operations, without entering into the demonftration of 

 the theorems themfelves. 



Proi'. XII. 



Every indeterminate equation of the fecond degree falls 

 under the general formula 



a.t' + b xy + cy- + dx + cy + / = o, 



and this may always he transformed to the more fimple form 



u- - A/'= B. 



The method of performing this transformation will be feen 

 from the following partial example, and the formulas we have 

 given it being univerfally the fame in all cafes. 



Affume id — 3 a e = g ; d — i^a f ::= h ; [by + dy 

 - 4 a (<■/ + /■_,;+ /) = t; b- j^ac = A; Ay + g 

 = u;g-Ah^ B. 



Then it will be found, by the deyelopement of thefe ex- 

 preffions, that 



"x' + bxy + i-/ + dx + ey +/= i/= — A /■■■ — B = Q, 

 becomes a — A/'^= B. 



And having found the values of u and / in this laft equa- 

 tion, thofe of .r and y are readily derived in the equation 

 propofed. 



F — ^is2^^ A — LuAnIi ■ 



■^ ^ A ' ■' ~ 2 a ' 



or, fubftituting for y in the laft, we have the following 



- S 

 dj (b- 



b — ^ac) 



Ex. I. — Transform the equation 



3 A' f 8.Vjr _3J,*+ 2.V -5> = 

 I its fi.npleftform. 



Here a — i, b — 9, c ^ — I, d — 2, . 

 ■ I ic. 



Whence C f — J^ac =■ A := 100 



\bd— 2ac— v:= 46 



) d - 4"./'"= ^ = 'i=4 



(.■-A// = B= - 130284 



5./ = 



and 



