Addition to the Memoir on Tschirnhausen's Transformation

VI. Addition to the Memoir on Tschirnhausen's Transformation. By Arthur Cayley, F.R.S. Received October 24,—Read December 7, 1866. In the memoir "On Tschirnhausen's Transformation," Philosophical Transactions, vol. clii. (1862) pp. 561–568, I considered the case of a quartic equation: viz. it was shown that the equation \[(a, b, c, d, e)(x, 1)^4 = 0\] is, by the substitution \[y = (ax + b)B + (ax^2 + 4bx + 3c)C + (ax^3 + 4bx^2 + 6cx + 3d)D,\] transformed into \[(1, 0, C, D, E)(y, 1)^4 = 0,\] where \((C, D, E)\) have certain given values. It was further remarked that \((C, D, E)\) were expressible in terms of \(U', H', \Phi'\), invariants of the two forms \((a, b, c, d, e)(X, Y)^4\), \((B, C, D)(Y, -X)^2\), of \(I, J\), the invariants of the first, and of \(\Theta'\), \(= BD - C^2\), the invariant of the second of these two forms,—viz. that we have \[C = 6H' - 2I\Theta',\] \[D = 4\Phi',\] \[E = IU'^3 - 3H'^2 + I^2\Theta'^2 + 12J'\Theta'U' + 2I'\Theta'H'.\] And by means of these I obtained an expression for the quadrinvariant of the form \[(1, 0, C, D, E)(y, 1)^4;\] viz. this was found to be \[= IU'^2 + \frac{4}{3}I^2\Theta'^2 + 12J\Theta'U'.\] But I did not obtain an expression for the cubinvariant of the same function: such expression, it was remarked, would contain the square of the invariant \(\Phi'\); it was probable that there existed an identical equation, \[JU'^2 - IU'^2H' + 4H'^3 + M\Theta' = - \Phi'^2,\] which would serve to express \(\Phi'^2\) in terms of the other invariants; but, assuming that such an equation existed, the form of the factor \(M\) remained to be ascertained; and until this was done, the expression for the cubinvariant could not be obtained in its most simple form. I have recently verified the existence of the identical equation just referred to, and have obtained the expression for the factor \(M\); and with the assistance of this identical equation I have obtained the expression for the cubinvariant of the form \[(1, 0, C, D, E)(y, 1)^4.\] The expression for the quadrinvariant was, as already mentioned, given in the former memoir: I find that the two invariants are in fact the invariants of a certain linear function of $U, H$; viz. the linear function is $= U'U + \frac{2}{3} \Theta'H$; so that, denoting by $I^*$, $J^*$, the quadrinvariant and the cubinvariant respectively of the form $$ (1, 0, C, D, G(x, y), 1)^4, $$ we have $$ I^* = \tilde{I}(U'U + 4\Theta'H), $$ $$ J^* = \tilde{J}(U'U + 4\Theta'H), $$ where $\tilde{I}, \tilde{J}$ signify the functional operations of forming the two invariants respectively. The function $(1, 0, C, D, G(x, y), 1)^4$, obtained by the application of Tschirnhausen's transformation to the equation $$ (a, b, c, d, e(x, 1)^4 = 0, $$ has thus the same invariants with the function $$ U'U + 4\Theta'H = U'(a, b, c, d, e(x, 1)^4 + 4\Theta'(ac - b^2, ad - bc, ae + 2bd - 3c^2, be - cd, ce - d^2(x, 1)^4, $$ and it is consequently a linear transformation of the last-mentioned function; so that the application of Tschirnhausen's transformation to the equation $U = 0$ gives an equation linearly transformable into, and thus virtually equivalent to, the equation $$ U'U + 4\Theta'H = 0, $$ which is an equation involving the single parameter $\frac{4\Theta'}{U'}$: this appears to me a result of considerable interest. It is to be remarked that Tschirnhausen's transformation, wherein $y$ is put equal to a rational and integral function of the order $n - 1$ (if $n$ be the order of the equation in $x$), is not really less general than the transformation wherein $y$ is put equal to any rational function $\frac{V}{W}$ whatever of $x$; such rational function may, in fact, by means of the given equation in $x$, be reduced to a rational and integral function of the order $n - 1$; hence in the present case, taking $V, W$ to be respectively of the order $n - 1, = 3$, it follows that the equation in $y$ obtained by the elimination of $x$ from the equations $$ (a, b, c, d, e(x, 1)^4 = 0, $$ $$ y = \frac{(\alpha, \beta, \gamma, \delta(x, 1)^3}{(\alpha', \beta', \gamma', \delta'(x, 1)^3} $$ is a mere linear transformation of the equation $AU + BH = 0$, where $A, B$ are functions (not as yet calculated) of $(a, b, c, d, e, \alpha, \beta, \gamma, \delta, \alpha', \beta', \gamma', \delta')$. Article Nos. 1, 2, 3.—Investigation of the identical equation $$ JU'^3 - IU'^2H' + 4H'^2 + M\Theta' = -\Phi'^2. $$ 1. It is only necessary to show that we have such an equation, $M$ being an invariant, in the particular case \(a = e = 1\), \(b = d = 0\), \(c = \theta\), that is for the quartic function \((1, 0, \theta, 0, 1)\); for, this being so, the equation will be true in general. Writing the equation in the form \[-M\Theta' = U'^2(JU' - IH') + 4H'^3 + \Phi'^2,\] and observing that we have \[U' = (B^2 + D^2) + 2\theta BD + 4\theta C^2,\] \[H' = \theta(B^2 + D^2) + (1 + \theta^2)BD - 4\theta^2 C^2,\] \[\Theta' = BD - C^2,\] \[\Phi' = (1 - 9\theta^2)C(B^2 - D^2),\] \[I = 1 + 3\theta^2,\] \[J = \theta - \theta^3,\] and thence \[JU' - IH' = -4\theta(B^2 + D^2) + (-1 - 2\theta^2 - 5\theta^4)BD + (8\theta^2 + 8\theta^4)C^2,\] the equation becomes \[-(BD - C^2)M =\] \[-4\theta(B^2 + D^2) + (-1 - 2\theta^2 - 5\theta^4)BD + (8\theta^2 + 8\theta^4)C^2\] \[\times \{B^2 + D^2 + 2\theta BD + 4\theta C^2\}^2\] \[+ 4\{\theta(B^2 + D^2) + (1 + \theta^2)BD - 4\theta^2 C^2\}^3\] \[+(1 - 9\theta^2)^2 C^2 \{(B^2 + D^2)^2 - 4B^2D^2\}.\] 2. It is found by developing that the right-hand side is in fact divisible by \(BD - C^2\), and that the quotient is \[-(-1 + 10\theta^2 - 9\theta^4)(B^2 + D^2)^2\] \[+(8\theta + 16\theta^3 - 24\theta^5)(B^2 + D^2)BD\] \[+(4 + 8\theta^2 + 4\theta^4 - 16\theta^6)B^2D^2\] \[+(-64\theta^3 - 192\theta^5)(B^2 + D^2)C^2\] \[+(16\theta^2 - 416\theta^4 - 112\theta^6)BDC^2\] \[+(-128\theta^4 + 128\theta^6)C^4.\] 3. This is found to be \[-I^2U'^2 + 12JU'H' + 4IH'^2\] \[-8IJU'\Theta'\] \[-16J^2\Theta'^2,\] which is consequently the value of \(-M\). We have therefore \[-\Phi'^2 = JU'^3 - IU'^2H' + 4H'^3\] \[-(I^2U'^2 - 12JU'H' - 4IH'^2)\Theta'\] \[-8IJU'\Theta'^2\] \[-16J^2\Theta'^3,\] which is the required identical equation. Article No. 4.—Calculation of the Cubinvariant. 4. We have \[ J^* = \frac{1}{6} C \cdot G - \left( \frac{1}{6} C \right)^3 - \left( \frac{1}{4} D \right)^2 \] \[ = (H - \frac{1}{3} I \Theta') \{ IU'^2 - 3 H'^2 + (12 JU' + 2 IH') \Theta' + I^2 \Theta'^2 \} \] \[ - (H - \frac{1}{3} I \Theta')^3 \] \[ - \Phi'^2, \] whence, substituting for \( -\Phi'^2 \) its value and reducing, we find \[ J^* = IU'^2 + \Theta' \cdot \frac{2}{3} I^2 U'^2 + \Theta'^2 (4IJU') + \Theta'^3 (16J^2 - \frac{8}{7} I^3). \] Article No. 5.—Final expressions of the two Invariants. The value of \( I^* \) has been already mentioned to be \( I^* = IU'^2 + \Theta' 12JU' + \Theta'^2 \cdot \frac{4}{3} I^2 \), and it hence appears that the values of the two invariants may be written \[ I^* = (I, 18J, 3I^2 \chi U', \frac{2}{3} \Theta')^2, \] \[ J^* = (J, I^2, 9IJ, -I^3 + 54J^2 \chi U', \frac{2}{3} \Theta')^3. \] But we have (see Table No. 72 in my "Seventh Memoir on Quantics"†) \[ \tilde{I}(\alpha U + 6 \beta H) = (I, 18J, 3I \chi \alpha, \beta)^2 \] \[ \tilde{J}(\alpha U + 6 \beta H) = (J, I^2, 9IJ, -I^3 + 54J^2 \chi \alpha, \beta)^3; \] so that, writing \( \alpha = U', \beta = \frac{2}{3} \Theta' \), we have \[ I^* = \tilde{I}(U'U + 4 \Theta'H), \] \[ J^* = \tilde{J}(U'U + 4 \Theta'H); \] or the function \((1, 0, C, D, G(y, 1)^4\) obtained from Tschirnhausen's transformation of the equation \(U = 0\) has the same invariants with the function \(U'U + 4 \Theta'H\); or, what is the same thing, the equation \((1, 0, C, D, G(y, 1)^4) = 0\) is a mere linear transformation of the equation \(U'U + 4 \Theta'H = 0\); which is the above-mentioned theorem. † Philosophical Transactions, vol. cl. (1861), pp. 277–292.