Elements of Algebra

534 ADDITIONS. CHAP. IV. and these last numbers $\beta$, and $\alpha$, will be the least values of $x$ and $y$. If the number of the remainders were odd, $g$ for instance being the last remainder $= 1$, then we must make $f \pm 1 = \zeta$, $\frac{\zeta e \mp 1}{f} = \varepsilon$, $\frac{\varepsilon d \pm 1}{c} = \delta$, &c. It is easy to see that this method is fundamentally the same as that of Chap. I.; but it is less convenient, because it requires divisions. Those who are curious in such speculations, will see with pleasure, in the work of M. Bachet, the artifices which he has employed to arrive at the foregoing Rule, and to deduce from it a complete solution of equations of the form, $ax - by = c$. CHAP. IV. *General method for resolving, in Integer Numbers, Equations with two unknown Quantities, of which one does not exceed the first Degree.* [APPENDIX TO CHAP. III.] 46. Let the general equation, $a+bx+cy+dx^2+exy+gy^2+fx^3+hx^4+kx^3y+$, &c. $= 0$ be proposed, in which the coefficients $a, b, c$, &c. are given integer numbers, and $x$ and $y$ two indeterminate numbers, which must also be integers. Deducing the value of $y$ from this equation, we shall have $y = - \frac{a+bx+dx^2+fx^3+hx^4+, \&c.}{c+ex+gx^2+kx^3+, \&c.}$ so that the question will be reduced to finding an integer number, which, when taken for $x$, makes the numerator of this fraction divisible by its denominator. Let us suppose $p = a + bx + dx^2 + fx^3 + hx^4+, \&c.$ $q = c + ex + gx^2 + kx^3 +, \&c.$ and taking $x$ out of both these equations by the ordinary rules of Algebra, we shall have a final equation of this form, $A + Bp + Cq + Dp^2 + Epq + Fq^2 + Gp^3 +, \&c. = 0,$ where the coefficients $A, B, C$, &c. will be rational and integer functions of the numbers $a, b, c$, &c.