CHAP. Xlll] Ax*+2Bxy+Cy*+2Dx+2Ey+F = 0. 417 



By the substitution p = ax+p, q = ay+y, we get ap 2 +2bpq+cq* = aA. The 

 theory of binary quadratic forms leads to all representations of A by the 

 form (a, b, c). From the resulting sets of values of x, y, discard those which 

 are not integral [cf. Smith 88 ]. 



To find (art. 300) the rational solutions of (11), set x = t/v, y = u/v, and 

 find the integral solutions of the resulting equation which is of the form 

 considered by Gauss. 147 



J. L. Wezel 86 reduced ax z +cxy+dx+ey+C = Q to xiy^ = k by a linear 

 substitution, and treated equations solvable rationally for one variable. 

 For ax 2 +by' 2 -i-2cxy + C = 0, we solve (p. 40) for x and find no trouble unless 

 B = c~ ab is positive and 4=D. The latter case is treated elegantly by 

 continued fractions. Develop the root r = (V# c)/o of az 2 +2cz-\-b = Q. 

 Let Q = (VB+7r)/C' be the complete quotient with denominator C, and 

 pfq the convergents immediately preceding this. Then 



ap 2 +2cpq+bq>=C, 



since V# is irrational. For ax z +by z -\-cxy - J rdx+ey+C = Q, we set 



x = (x'+2bd-ce}/D, y = (y'+2ae-cd)/D, 

 where D = c 2 4a&, and get an equation of the form last treated : 



E. Desmarest 87 noted that the substitution X = xfa reduces the solution 

 of aX 2 +bX+c = Ky to the problem to find multiples of a satisfying an 

 equation of type f x =x z -\-qx-\-r = Py. To solve a particular equation of the 

 latter type, he would employ two auxiliary doubly-entry tables, a com- 

 plicated method based upon the functions 



C.P^-2 =fnN 2 -f'nN+1, P 2 *-1 =fnN 2 +f' n N+ 1 



and the fact that their products by / are also of the form f x) where 

 x = f n N n q and f n N+n, respectively. One of the auxiliary tables has the 

 headings /, Pi, P 2) and in the body of the table are entered the values 

 for successive K's of the roots R and remainders p defined, for example, 

 when N = 2K-\-l, by use of 



P 2N = R*+p, R = (2K+2)n+qK-q-l, p = A(K+l) 2 , A = 4r-q\ 



Troublesome methods are indicated (pp. 42, 43) by means of which the 

 square R 2 nearest to the given P enables us to find the entry in the body of 

 the table which will yield the desired value of n such that the heading of the 

 column of the entry will for this n be the value of y (or a known multiple 

 of y). The example X 2 +31X+21=PY is treated (pp. 24-25, 301-2) 

 for all primes P< 1000; but he knew (p. 104) that it can be transformed by 

 X = x 15 into x 2 -\-x-\-l = Py, which is proved to be solvable if and only 

 if the prime P is 3 or 3<?+l. 



86 Annales Acad. Leodiensis, Liege, 1821-2, 1-48. 



87 The'orie des nombres. Traite" de 1'analyse inde"termmee du second degre" a deux inconnuea 



. . . , Paris, 1852, 4-126. 



28 



