Elements of Algebra

CHAP. IX. ADDITIONS. 593 ductions, which are not easily performed, we may set about it, if it be judged more convenient, in the following manner. Let us suppose, in general, $x + sy + s^2z + s^3t = \rho$; and, as $s$ is determined by the equation, $s^4 - as^3 + bs^2 - cs + d = 0$, let us take away $s$ from these two equations by the common rules, and the equation, which results, after expunging $s$, being arranged according to the unknown quantity $\rho$, will rise to the fourth degree; so that it may be put into this form, $\rho^4 - N\rho^3 + P\rho^2 - Q\rho + R = 0$. Now, the cause of this equation in $\rho$ rising to the fourth degree is, that $s$ may have the four values $\alpha$, $\beta$, $\gamma$, $\delta$; and also that $\rho$ may likewise have these four corresponding values, $x + \alpha y + \alpha^2 z + \alpha^3 t$ $x + \beta y + \beta^2 z + \beta^3 t$ $x + \gamma y + \gamma^2 z + \gamma^3 t$ $x + \delta y + \delta^2 z + \delta^3 t$, which are nothing but those factors, the product of which is required. Wherefore, since the last term $R$ must be the product of all the four roots, or values of $\rho$, it follows, that this quantity, $R$, will be the product required. But we have now said enough on this subject, which we might resume, perhaps, on some other occasion. I shall here close these Additions, which the limits I prescribed to myself will not permit me to carry any farther; perhaps they have already been found too long: but the subjects I have considered being rather new and little known, I thought it incumbent on me to enter into several details, necessary for the full illustration of the methods which I have explained, and of their different uses. THE END. Q Q