Elements of Algebra

CHAP. IX. ADDITIONS. 583 Let us, therefore, in the same manner, make $t = 2u - x$, and we shall have $-2u^2 + 8ux - 5x^2 = 1$; wherefore, &c. Continuing thus to take the limits always in plus, we shall never come to a transformed equation, in which the coefficient of the first term is equal to unity, which is necessary to our finding a solution of the equation proposed. The same thing must happen, whenever we take the first limit in minus, and all the succeeding in plus; the reason of this might be given a priori; but as the reader can easily deduce it from the principles of our theory, I shall not dwell on it. It is sufficient for the present to have shewn the necessity of investigating these problems more fully, and more rigorously, than has hitherto been done. *** CHAP. IX. Of the manner of finding Algebraic Functions of all Degrees, which, when multiplied together, may always produce Similar Functions. [APPENDIX TO CHAP. XI. AND XII.] 88. I believe I had, at the same time with M. Euler, the idea of employing the irrational, and even imaginary factors of formulæ of the second degree, in finding the conditions, which render those formulæ equal to squares, or to any powers. On this subject, I read a Memoir to the Academy in 1768, which has not been printed; but of which I have given a summary at the end of my researches on Indeterminate Problems, which are to be found in the volume for the year 1767, printed in 1769, before even the German edition of M. Euler's Algebra. In the place now quoted, I have shewn how the same method may be extended to formulæ of higher dimensions than the second; and I have by these means given the solution of some equations, which it would perhaps have been extremely difficult to resolve in any other way. It is here intended to generalise this method still more, as it seems to deserve the attention of mathematicians, from its novelty and singularity. 89. Let $\alpha$ and $\beta$ be the two roots of the quadratic equation, $s^2 - as + b = 0$, and let us consider the product of these two factors,