Elements of Algebra

592 ADDITIONS. CHAP. IX. $(x + \alpha y + \alpha^2 z)^m$, it will be found composed of three parts, one all rational, another all multiplied by $\alpha$, and the third all multiplied by $\alpha^2$; so that we shall only have to compare the first to $x$, the second to $\alpha y$, and the third to $\alpha^2 z$, and, by these means, we shall have $X = P + P'C' + P''C'' + P'''C''' + P^{iv}C^{iv}, \ldots$ $Y = -P'B' - P''B'' - P'''B''' - P^{iv}B^{iv}, \ldots$ $Z = P'A' + P''A'' + P'''A''' + P^{iv}A^{iv}, \ldots$ These values, therefore, will satisfy the equation, $x + \alpha y + \alpha^2 z = (x + \alpha y + \alpha^2 z)^m$; and as the root $\alpha$ does not enter into the expressions of $x$, $y$, and $z$, it is evident, that we may change $\alpha$ into $\beta$, or into $\gamma$; so that we shall have both $x + \beta y + \beta^2 z = (x + \beta y + \beta^2 z)^m$, and $x + \gamma y + \gamma^2 z = (x + \gamma y + \gamma^2 z)^m$. If we now multiply these three equations together, it is evident, that the first member will be the same as that of the given equation, and that the second will be equal to a power, $m$, the root of which being called $v$, we shall have $v = x^3 + ax^2y + (a^2 - 2b)x^2z + bxy^2 + (ab - 3c)xyz + (b^2 - 2ac)xz^2 + cy^3 + acy^2z + bcyz^2 + c^2z^3$. Thus, we shall have the values required of $x, y, z$, and $v$, which will contain three indeterminates, $x, y, z$. 94. If we wished to find formulae of four dimensions, having the same properties as those we have now examined, it would be necessary to consider the product of four factors of this form, $x + \alpha y + \alpha^2 z + \alpha^3 t$ $x + \beta y + \beta^2 z + \beta^3 t$ $x + \gamma y + \gamma^2 z + \gamma^3 t$ $x + \delta y + \delta^2 z + \delta^3 t$, supposing $\alpha, \beta, \gamma, \delta$ to be the roots of a biquadratic equation, such as $s^4 - as^3 + bs^2 - cs + d = 0$; we shall thus have $\alpha + \beta + \gamma + \delta = a$, $\alpha\beta + \alpha\gamma + \alpha\delta + \beta\gamma + \beta\delta + \gamma\delta = b$, $\alpha\beta\gamma + \alpha\beta\delta + \alpha\gamma\delta + \beta\gamma\delta = c$, $\alpha\beta\gamma\delta = d$, by which means we may determine all the coefficients of the different terms of the product in question, without knowing the roots $\alpha, \beta, \gamma, \delta$. But as this requires different re-