INDETERMINATE ANALYSIS. 



end y =: arr. — q, and it is o'jvious that this fubililution will 

 give 



„ {bm-p) -b [am -q) = - i ; 

 and here, by means of the ind terminate m an indefinite num- 

 ber of values of .■<; andy may be determined. 



If from one known cafe, as «/> — iy =: + i, the general 

 values of a; and^' in the fame equation ax — by = -f i were 

 required j we fhould only have to make x =^ mb + p, and 

 y ::^ m a + q, and we fliculd have dill 



a {mi + p) — b {ma + q) — ap — bq = + I, 

 where, by means of the indeterminate m, Jin indefinite number 

 of values of .\- aiidj' may be obtained. We will now illullrate 

 wliat has been taught by an example, -and then proceed to 

 the more general equations ; each of which, however, will 

 be found to depend upon the one we have been confidering. 



Ex. 3. — Find the general values of .r and y, in the equa- 

 tion 



1)6(6 

 6 



Quotients i, 2, 6 



r I ; IQ 



Fraaions \ -, -, — 



lehave therefore/ = 3, and(7 = 2 ; and this gives 



Therefore the general values of x and v are 



X = 19 w + /, and y = 13 w + ?. O"" 

 x= igm + 3, and ^1= 13 m + 2. 



AlTuming therefore m = 0,1, 2, 3, 4, &c. ; we have the 



following values of 



' and 



m = o, i, 2, 3, 4, 5 &c. 

 * — 3, 22, 41, 60, 79, 98 &c. 

 y = 2, I3-, 28, 41, 54, 67 &C. 

 ■which feries may be continued at p'eafure. 



If the propofcd equation had been 13 .v — igy= — I. 

 then having found p — S' ^"^ ? = 2» as above ; we mull 

 have made 



x= ignt — 3; ind y— 13m — 2 ; 



and then, by affuming m as before, we (hould have 



ffi = o. I. 2. 3. 4' 5 &c- 



x = - 3, 16, 3,-, 54, 73, 92 &c. 



J, = - 2, J I, 24, 37, 50, 63 &c. 



where it may be obfervcd, that the fuccefiivc values of x 



-ai.d y, in both cafos, form a feries of arithmeticals, and may 



tlierefore be continued with great facility. 



Pjiop. II. 

 "To find tiie general values of x and y, in the equation 



ax — by = ± c. 

 In tlie firft place ive muft have either a and b prime to each 

 other, or if they have a common mcafure, c mt:ft have the 

 /ame, for olherwife the equation will be impoflil Ij ; and iu 

 this latter cafe, tlie whole equation may be divided by that 

 Asmiuon mcafure, and thus reduced to one in which a and b 

 4 



are prime to each other: it will, therefore, only beneci :' 

 to confider the quantities a and b as prime to each o; 

 Alfo, after what has been taught in the foregoing prujj 

 tion, we may always fuppofe that we know the caie 

 ap — b g = + J ; it will therefore be fufHcient in tlr.i 

 p^ace to ihew how the general values of .\- and^, in the equa- 

 tion ax — by — + c, may be deduced from the known cale 

 ap — bq - ±1. 



In the firft place it is obvious, that fince 

 ap - bq= + 1, 

 we flial! have 



acp — hcq=: +c; 

 but this furnid.es only one folution, and in order to have tlir 

 general values of xsxidy, we muft fubilitute x=-mb^cp; 

 and y = ma +cq ; which give 



c{mb±,p) - b{ma ± c q) = ± c ; 

 the ambiguous Cgn +, in the two values of *■ and ji, being -f- 

 when ap — bq has the fame fign with c, but — when it 

 has a contrary one. 



Ex. I. — Find the values of x and ji, in the equation 



9*- 13^ = 1°. 

 Firft, in the equation 9/ — 13 9 = + i. we have/ = 3, 

 and q ^ 1, which gives 9/ — 13 ^ = -(- i ; and this being 

 the lame fign with I o in the propofed equation, tlie general 

 values of jT and^ are 



r.v=l3m + 3c,or , $y = ()m + 2c,0T 

 \x:^lim + 30; 13r = 9m + 20 



And by afi"uming here m =: — 2, — 1,0, I, 2, &c. we 

 have the following values of x and^ : 



rj = — 2, — I, o, I, 2, 3 &c, 



J' = 4. 17' 30' 43. 5^. 69 &c. 



J— 2, II, 20, 29, 38, 47 &c. 



each of which values has the required conditions, for 



9. 4 - 13 • 2 = 10 



9. 17 - 13 . II = 10 



9 . 30 — 13 . 20 = 10 



9 • 43 - 13 • 29 =^ 10 



&c. &c. &c. 



Ex. 2. — Find the values of x and j, in the equation 

 7*- I2>= 19- 



and(; 



Firft in the equation 7/ 



3; 



129 



has a different 



- I, we have/ = ^, 

 ilccn from 10 in the 



propofed equation ; therefore the general values of x and y 



are 



fx=l2m-5..9; J f^=7m_3.l9.or 

 \x=.i2m- 95; |j, = 7m- 57 



where, by taking m = 9, 10, 1 1, &c. in order that x and_)P 

 may be pofitive, we have 



•■< = 13. 2J, 37, 49, 61, 73, 8j &c. 



y — 6, 13, 20, 27, 34, 41, 48 &c. 

 and in a fimilar manner may any poffible equation ax — by 

 = + c be refolved. 



Prop. III. 

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



ax + by ^= c. 



In the foregoing propofition, where the difference of two 



quantities was the fubjeft of confideration, we found that the 



number of folutions was infinite, provided that a and b were 



prime to each other ; but in confidering the fum of two 



quantities, 



