{
  "id": "b73e17279d037c6cfc187df2f8432d8abe6a42a8",
  "text": "X. On the Sextactic Points of a Plane Curve. By A. Cayley, F.R.S.\n\nReceived November 5,—Read December 22, 1864.\n\nIt is, in my memoir \"On the Conic of Five-pointic Contact at any point of a Plane Curve\"*, remarked that as in a plane curve there are certain singular points, viz. the points of inflexion, where three consecutive points lie in a line, so there are singular points where six consecutive points of the curve lie in a conic; and such a singular point is there termed a \"sextactic point.\" The memoir in question (here cited as \"former memoir\") contains the theory of the sextactic points of a cubic curve; but it is only recently that I have succeeded in establishing the theory for a curve of the order $m$. The result arrived at is that the number of sextactic points is $= m(12m - 27)$, the points in question being the intersections of the curve $m$ with a curve of the order $12m - 27$, the equation of which is\n\n$$\\left(12m^2 - 54m + 57\\right)H \\text{ Jac. } (U, H, \\Omega_{\\bar{\\nu}})$$\n\n$$+ (m-2)(12m-27)H \\text{ Jac. } (U, H, \\Omega_{\\bar{\\nu}})$$\n\n$$+ 40(m-2)^2 \\text{ Jac. } (U, H, \\Psi) = 0,$$\n\nwhere $U = 0$ is the equation of the given curve of the order $m$, $H$ is the Hessian or determinant formed with the second differential coefficients $(a, b, c, f', g, h)$ of $U$, and, $(A, B, C, F, G, H)$ being the inverse coefficients ($A = bc - f'^2$, &c.), then\n\n$$\\Omega = (A, B, C, F, G, H)(\\partial_x, \\partial_y, \\partial_z)^3H,$$\n\n$$\\Psi = (A, B, C, F, G, H)(\\partial_xH, \\partial_yH, \\partial_zH)^3;$$\n\nand Jac. denotes the Jacobian or functional determinant, viz.\n\n$$\\text{Jac. } (U, H, \\Psi) = \\begin{vmatrix}\n\\partial_xU, \\partial_yU, \\partial_zU \\\\\n\\partial_xH, \\partial_yH, \\partial_zH \\\\\n\\partial_x\\Psi, \\partial_y\\Psi, \\partial_z\\Psi\n\\end{vmatrix},$$\n\nand Jac. $(U, H, \\Omega)$ would of course denote the like derivative of $(U, H, \\Omega)$; the subscripts $(\\bar{\\nu}, \\bar{\\nu})$ of $\\Omega$ denote restrictions in regard to the differentiation of this function, viz. treating $\\Omega$ as a function of $U$ and $H$,\n\n$$\\Omega = (A, B, C, F, G, H)(a', b', c', f', 2f', 2g', 2h'),$$\n\nif $(a', b', c', f', g', h')$ are the second differential coefficients of $H$, then we have\n\n$$\\partial_x\\Omega = (\\partial_xA, \\ldots \\times a', \\ldots) \\quad (= \\partial_x\\Omega_{\\bar{\\nu}})$$\n\n$$+ (\\ A, \\ldots \\times \\partial_xa', \\ldots) \\quad (= \\partial_x\\Omega_{\\bar{\\nu}});$$\n\n* Philosophical Transactions, vol. cxlix. (1859) pp. 371—400.\nviz. in $\\partial_x \\Omega_H$ we consider as exempt from differentiation $(a', b', c', f', g', h')$ which depend upon $H$, and in $\\partial_x \\Omega_U$ we consider as exempt from differentiation $(A, B, C, F, G, H)$ which depend upon $U$. We have similarly\n\n$$\\partial_y \\Omega = \\partial_y \\Omega_H + \\partial_y \\Omega_U,$$\n\nand in like manner\n\n$$\\partial_z \\Omega = \\partial_z \\Omega_H + \\partial_z \\Omega_U;$$\n\nand in like manner\n\n$$\\text{Jac.}(U, H, \\Omega) = \\text{Jac.}(U, H, \\Omega_H) + \\text{Jac.}(U, H, \\Omega_U),$$\n\nwhich explains the signification of the notations $\\text{Jac.}(U, H, \\Omega_H), \\text{Jac.}(U, H, \\Omega_U)$.\n\nThe condition for a sextactic point is in the first instance obtained in a form involving the arbitrary coefficients $(\\lambda, \\mu, v);$ viz. we have an equation of the order 5 in $(\\lambda, \\mu, v)$ and of the order $12m-22$ in the coordinates $(x, y, z)$. But writing $\\Omega = \\lambda x + \\mu y + vz$, by successive transformations we throw out the factors $\\Omega^2, \\Omega, \\Omega, \\Omega$, thus arriving at a result independent of $(\\lambda, \\mu, v);$ viz. this is the before-mentioned equation of the order $12m-27$. The difficulty of the investigation consists in obtaining the transformations by means of which the equation in its original form is thus divested of these irrelevant factors.\n\nArticle Nos. 1 to 6.—Investigation of the Condition for a Sextactic Point.\n\n1. Following the course of investigation in my former memoir, I take $(X, Y, Z)$ as current coordinates, and I write\n\n$$Y = (*X, Y, Z)^m = 0$$\n\nfor the equation of the given curve; $(x, y, z)$ are the coordinates of a particular point on the given curve, viz. the sextactic point; and $U, = (*x, y, z)^m$, is what $Y$ becomes when $(x, y, z)$ are written in place of $(X, Y, Z);$ we have thus $U = 0$ as a condition satisfied by the coordinates of the point in question.\n\n2. Writing for shortness\n\n$$DU = (X \\partial_x + Y \\partial_y + Z \\partial_z) U,$$\n\n$$D^2U = (X \\partial_x + Y \\partial_y + Z \\partial_z)^2 U,$$\n\nand taking $\\Pi = aX + bY + cZ = 0$ for the equation of an arbitrary line, the equation\n\n$$D^2U - \\Pi DU = 0$$\n\nis that of a conic having an ordinary (two-pointic) contact with the curve at the point $(x, y, z);$ and the coefficients of $\\Pi$ are in the former memoir determined so that the contact may be a five-pointic one; the value obtained for $\\Pi$ is\n\n$$\\Pi = \\frac{3}{H} DH + \\Lambda DU,$$\n\nwhere\n\n$$\\Lambda = \\frac{1}{9H^3}(-3\\Omega H + 4\\Psi).$$\n\n3. This result was obtained by considering the coordinates of a point of the curve as functions of a single arbitrary parameter, and taking\n\n$$x + dx + \\frac{1}{2}d^2x + \\frac{1}{6}d^3x + \\frac{1}{24}d^4x, y + &c., z + &c.$$\nfor the coordinates of a point consecutive to \\((x, y, z)\\); for the present purpose we must go a step further, and write for the coordinates\n\n\\[\n\\begin{align*}\nx + dx + \\frac{1}{2}d^2x + \\frac{1}{6}d^3x + \\frac{1}{24}d^4x + \\frac{1}{120}d^5x, \\\\\ny + dy + \\frac{1}{2}d^2y + \\frac{1}{6}d^3y + \\frac{1}{24}d^4y + \\frac{1}{120}d^5y, \\\\\nz + dz + \\frac{1}{2}d^2z + \\frac{1}{6}d^3z + \\frac{1}{24}d^4z + \\frac{1}{120}d^5z.\n\\end{align*}\n\\]\n\n4. Hence if\n\n\\[\n\\partial_1 = dx\\partial_x + dy\\partial_y + dz\\partial_z, \\quad \\partial_2 = d^2x\\partial_x + d^2y\\partial_y + d^2z\\partial_z, \\quad \\text{etc.},\n\\]\n\nwe have, in addition to the equations\n\n\\[\n\\begin{align*}\nU &= 0, \\\\\n\\partial_1 U &= 0, \\\\\n(\\partial_1^2 + 2\\partial_2)U &= 0, \\\\\n(\\partial_1^3 + 3\\partial_1\\partial_2 + \\partial_3)U &= 0, \\\\\n(\\partial_1^4 + 6\\partial_1^2\\partial_2 + 4\\partial_1\\partial_3 + 3\\partial_2^2 + \\partial_4)U &= 0,\n\\end{align*}\n\\]\n\nof my former memoir, the new equation\n\n\\[\n(\\partial_1^3 + 10\\partial_1^2\\partial_2 + 10\\partial_1\\partial_3 + 15\\partial_2^2 + 5\\partial_1\\partial_4 + 10\\partial_2\\partial_3 + \\partial_5)U = 0,\n\\]\n\nand in addition to the equations, \\((P = ax + by + cz)\\),\n\n\\[\n\\begin{align*}\n-(m-2)\\partial_1^2U + P\\cdot\\frac{1}{2}\\partial_1^3U &= 0 \\\\\n-\\frac{1}{3}[(m-1)\\partial_1^3 + 3(m-2)\\partial_1\\partial_2]U + P\\cdot\\frac{1}{6}(\\partial_1^3 + 3\\partial_1\\partial_2)U + \\partial_1P\\cdot\\frac{1}{2}\\partial_1^2U &= 0, \\\\\n-\\frac{1}{12}[(m-1)(\\partial_1^4 + 6\\partial_1^2\\partial_2) + (m-2)(4\\partial_1\\partial_3 + 3\\partial_2^2)]U \\\\\n+ P\\cdot\\frac{1}{24}(\\partial_1^4 + 6\\partial_1^2\\partial_2 + 4\\partial_1\\partial_3 + 3\\partial_2^2)U + \\partial_1P\\cdot\\frac{1}{6}(\\partial_1^3 + 3\\partial_1\\partial_2)U + \\frac{1}{2}\\partial_2P\\cdot\\frac{1}{3}\\partial_1^2U &= 0,\n\\end{align*}\n\\]\n\ngiving in the first instance\n\n\\[\nP = 2(m-2),\n\\]\n\n\\[\n\\partial_1P = \\frac{3}{2}\\frac{\\partial_1^3U}{\\partial_1^2U},\n\\]\n\n\\[\n\\partial_2P = \\frac{1}{2}\\left(\\frac{\\partial_1^4 + 6\\partial_1^2\\partial_2}{\\partial_1^2U} - \\frac{4}{9}\\frac{\\partial_1^3U}{\\partial_1^2U}(\\partial_1^3 + 3\\partial_1\\partial_2)\\right)U,\n\\]\n\nand leading ultimately to the before-mentioned value of \\(P\\), we have the new equation\n\n\\[\n-\\frac{1}{60}[(m-1)(\\partial_1^3 + 10\\partial_1^2\\partial_2 + 10\\partial_1\\partial_3 + 15\\partial_2^2) + (m-2)(5\\partial_1\\partial_4 + 10\\partial_2\\partial_3)]U \\\\\n+ P\\cdot\\frac{1}{120}(\\partial_1^5 + 10\\partial_1^3\\partial_2 + 10\\partial_1^2\\partial_3 + 15\\partial_1\\partial_2^2 + 5\\partial_1\\partial_4 + 10\\partial_2\\partial_3)U \\\\\n+ \\partial_1P\\cdot\\frac{1}{24}(\\partial_1^4 + 6\\partial_1^2\\partial_2 + 4\\partial_1\\partial_3 + 3\\partial_2^2)U \\\\\n+ \\frac{1}{3}\\partial_2P\\cdot\\frac{1}{6}(\\partial_1^3 + 3\\partial_1\\partial_2)U \\\\\n+ \\frac{1}{6}\\partial_3P\\cdot\\frac{1}{2}\\partial_1^2U = 0.\n\\]\n5. This may be written in the form\n\n\\[ -2[(m-1)(\\partial_1^5 + 10\\partial_1^3\\partial_2 + 10\\partial_1^2\\partial_3 + 15\\partial_1\\partial_2^2) + (m-2)(5\\partial_1\\partial_4 + 10\\partial_2\\partial_3)]U \\]\n\n\\[ + P(\\partial_1^5 + 10\\partial_1^3\\partial_2 + 10\\partial_1^2\\partial_3 + 15\\partial_1\\partial_2^2 + 5\\partial_1\\partial_4 + 10\\partial_2\\partial_3)U \\]\n\n\\[ + 5\\partial_1P(\\partial_1^4 + 6\\partial_1^2\\partial_2 + 4\\partial_1\\partial_3 + 3\\partial_2^2)U \\]\n\n\\[ + 10\\partial_2P(\\partial_1^3 + 3\\partial_1\\partial_2)U \\]\n\n\\[ + 10\\partial_3P(\\partial_1^2U) = 0; \\]\n\nor putting for \\( P \\) its value, \\( = 2(m-2) \\), the equation becomes\n\n\\[ -2(\\partial_1^5 + 10\\partial_1^3\\partial_2 + 10\\partial_1^2\\partial_3 + 15\\partial_1\\partial_2^2)U \\]\n\n\\[ + 5\\partial_1P(\\partial_1^4 + 6\\partial_1^2\\partial_2 + 4\\partial_1\\partial_3 + 3\\partial_2^2)U \\]\n\n\\[ + 10\\partial_2P(\\partial_1^3 + 3\\partial_1\\partial_2)U \\]\n\n\\[ + 10\\partial_3P.\\partial_2^2U = 0; \\]\n\nor, as this may also be written,\n\n\\[ 2(\\partial_1^5 + 10\\partial_1^3\\partial_2 + 10\\partial_1^2\\partial_3 + 15\\partial_1\\partial_2^2)U \\]\n\n\\[ + 5\\partial_1P.\\partial_4U + 10\\partial_2P.\\partial_3U + 10\\partial_3P.\\partial_2U = 0. \\]\n\n6. But the equation\n\n\\[ \\Pi = \\frac{2}{3} \\frac{1}{H} DH + \\Lambda DU, \\]\n\nwhich is an identity in regard to \\((X, Y, Z)\\), gives\n\n\\[ \\partial_1P = \\frac{2}{3} \\frac{1}{H} \\partial_1H, \\]\n\n\\[ \\partial_2P = \\frac{2}{3} \\frac{1}{H} \\partial_2H + \\Lambda \\partial_2U, \\]\n\n\\[ \\partial_3P = \\frac{2}{3} \\frac{1}{H} \\partial_3H + \\Lambda \\partial_3U; \\]\n\nand substituting these values, the foregoing equation becomes\n\n\\[ 2(\\partial_1^5 + 10\\partial_1^3\\partial_2 + 10\\partial_1^2\\partial_3 + 15\\partial_1\\partial_2^2)U \\]\n\n\\[ + (5\\partial_4U\\partial_1H + 10\\partial_3U\\partial_2H + 10\\partial_2U\\partial_3H) \\frac{2}{3} \\frac{1}{H} + \\Lambda .20\\partial_2U\\partial_3U = 0; \\]\n\nor putting for \\( \\Lambda \\) its value, \\( = \\frac{1}{9H^3}(-3\\Omega H + 4\\Psi) \\), and multiplying by \\( \\frac{9}{2}H^2 \\) this is\n\n\\[ 9H^2(\\partial_1^5 + 10\\partial_1^3\\partial_2 + 10\\partial_1^2\\partial_3 + 15\\partial_1\\partial_2^2)U \\]\n\n\\[ + 15H(\\partial_4U\\partial_1H + 2\\partial_3U\\partial_2H + 2\\partial_2U\\partial_3H) \\]\n\n\\[ + \\frac{1}{H}(-3\\Omega H + 4\\Psi).10\\partial_2U\\partial_3U = 0, \\]\n\nwhich is, in its original or unreduced form, the condition for a sextactic point.\nArticle Nos. 7 & 8.—Notations and Remarks.\n\n7. Writing, as in my former memoir, A, B, C for the first differential coefficients of U, we have \\( B_\\nu - C_\\mu, \\ C_\\lambda - A_\\nu, \\ A_\\mu - B_\\lambda \\) for the values of \\( dx, dy, dz \\), and instead of the symbol \\( D \\) used in my former memoir, I use indifferently the original symbol \\( \\partial_1 \\), or write instead thereof \\( \\partial \\), to denote the resulting value\n\n\\[\n\\partial_1 (= \\partial) = (B_\\nu - C_\\mu) \\partial_x + (C_\\lambda - A_\\nu) \\partial_y + (A_\\mu - B_\\lambda) \\partial_z,\n\\]\n\nand I remark here that for any function whatever \\( \\Omega \\), we have\n\n\\[\n\\partial \\Omega = \\begin{vmatrix}\nA, & B, & C \\\\\n\\lambda, & \\mu, & \\nu \\\\\n\\partial_x \\Omega, & \\partial_y \\Omega, & \\partial_z \\Omega\n\\end{vmatrix} = \\text{Jac.} (U, \\partial, \\Omega),\n\\]\n\nwhere \\( \\partial = \\lambda x + \\mu y + \\nu z \\). I write, as in the former memoir,\n\n\\[\n\\Phi = (\\mathcal{A}, \\mathcal{B}, \\mathcal{C}, \\mathcal{F}, \\mathcal{G}, \\mathcal{H})(\\lambda, \\mu, \\nu)^2;\n\\]\n\nand also\n\n\\[\n\\nabla = (\\mathcal{A}, \\mathcal{B}, \\mathcal{C}, \\mathcal{F}, \\mathcal{G}, \\mathcal{H})(\\lambda, \\mu, \\nu)(\\partial_x, \\partial_y, \\partial_z),\n\\]\n\nwhich new symbol \\( \\nabla \\) serves to express the functions \\( \\Pi, \\square \\), occurring in the former memoir; viz. we have \\( \\Pi = 2 \\nabla \\Phi, \\square = 2 \\nabla H \\), so that the symbols \\( \\Pi, \\square \\) are not any longer required.\n\n8. I remark that the symbols \\( \\partial, \\nabla \\) are each of them a linear function of \\( (\\partial_x, \\partial_y, \\partial_z) \\), with coefficients which are functions of the variables \\( (x, y, z) \\); and this being so, that for any function \\( \\Pi \\) whatever, we have\n\n\\[\n\\partial(\\nabla \\Pi) = (\\partial \\cdot \\nabla) \\Pi + \\partial \\nabla \\Pi,\n\\]\n\nviz. in \\( \\partial(\\nabla \\Pi) \\) we operate with \\( \\nabla \\) on \\( \\Pi \\), thereby obtaining \\( \\nabla \\Pi \\), and then with \\( \\partial \\) on \\( \\nabla \\Pi \\); in \\( (\\partial \\cdot \\nabla) \\Pi \\) we operate with \\( \\partial \\) upon \\( \\nabla \\) in so far as \\( \\nabla \\) is a function of \\( (x, y, z) \\), thus obtaining a new operating symbol \\( \\partial \\cdot \\nabla \\), a linear function of \\( (\\partial_x, \\partial_y, \\partial_z) \\), and then operate with \\( \\partial \\cdot \\nabla \\) upon \\( \\Pi \\); and lastly, in \\( \\partial \\nabla \\Pi \\), we simply multiply together \\( \\partial \\) and \\( \\nabla \\), thus obtaining a new operating symbol \\( \\partial \\nabla \\) of the form \\( (\\partial_x, \\partial_y, \\partial_z)^2 \\), and then operate therewith on \\( \\Pi \\); it is clear that, as regards the last-mentioned mode of combination, the symbols \\( \\partial \\) and \\( \\nabla \\) are convertible, or \\( \\partial \\nabla = \\nabla \\partial \\), that is, \\( \\partial \\nabla \\Pi = \\nabla \\partial \\Pi \\).\n\nIt is to be observed throughout the memoir that the point \\( (.) \\) is used (as above in \\( \\partial \\cdot \\nabla \\)) when an operation is performed upon a symbol of operation as operand; the mere apposition of two or more symbols of operation (as above in \\( \\partial \\nabla \\)) denotes that the symbols of operation are simply multiplied together; and when \\( \\partial \\nabla \\) is followed by a letter \\( \\Pi \\) denoting not a symbol of operation, but a mere function of the coordinates, that is in an expression such as \\( \\partial \\nabla \\Pi \\), the resulting operation \\( \\partial \\nabla \\) is performed upon \\( \\Pi \\) as operand; if instead of the single letter \\( \\Pi \\) we have a compound symbol such as \\( HU \\) or \\( H \\nabla \\Omega \\), so that the expression is \\( \\partial HU, \\partial H \\nabla \\Omega, \\partial \\nabla HU \\) or \\( \\partial \\nabla H \\nabla \\Omega \\), then it is to be understood that it is merely the immediately following function \\( H \\) which is operated upon by \\( \\partial \\) or \\( \\partial \\nabla \\); in the few instances where any ambiguity might arise a special explanation is given.\nArticle Nos. 9 to 11.—First Transformation.\n\n9. We have, assuming always \\( U = 0 \\), the following formulæ (see post, Article Nos. 31 to 33):\n\n\\[\n(\\partial_1^2 + 10\\partial_2^2 + 10\\partial_3^2 + 15\\partial_4^2)U \\\\\n= \\frac{s^2}{(m-1)^4} \\{ (27m^2 - 96m + 81)H\\Phi + (17m^2 - 56m + 51)\\Phi H \\} \\\\\n+ \\frac{s^3}{(m-1)^4} \\{ (-14m - 22)(\\partial \\cdot \\nabla)H - (10m - 18)\\partial \\nabla H \\} \\\\\n+ \\frac{s^4}{(m-1)^4} \\{ \\partial \\Omega \\},\n\\]\n\n\\[\n\\partial_4 U\\partial_1 H + 2\\partial_3 U\\partial_2 H + 2\\partial_2 U\\partial_3 H \\\\\n= \\frac{s^2}{(m-1)^4} \\{ (-6m^2 + 18m - 12)H^2\\Phi + (-17m^2 + 60m - 55)H\\Phi \\Phi \\} \\\\\n+ \\frac{s^3}{(m-1)^4} \\{ (2m - 2)H(\\partial \\cdot \\nabla)H + (8m - 16)\\partial \\nabla H \\} \\\\\n+ \\frac{s^4}{(m-1)^4} \\{ -\\Omega \\partial H \\},\n\\]\n\n\\[\n\\partial_3 U\\partial_3 U = \\frac{s^4}{(m-1)^4} H\\partial H.\n\\]\n\n10. And by means of these the condition becomes\n\n\\[\n0 = \\frac{s^2 H^2}{(m-1)^4} \\{ (153m^2 - 594m + 549)H\\Phi + (-102m^2 + 396m + 366)\\Phi H \\} \\\\\n+ \\frac{s^3 H}{(m-1)^4} \\{ (-96m + 168)H(\\partial \\cdot \\nabla)H + (-90m + 162)H\\partial \\nabla H + (120m - 240)\\partial \\nabla H \\} \\\\\n+ \\frac{s^4}{(m-1)^4} \\{ 9H^2\\Omega - 45H\\Omega \\partial H + 40\\Psi \\partial H \\},\n\\]\n\nbeing, as already remarked, of the degree 5 in the arbitrary coefficients \\((\\lambda, \\mu, \\nu)\\), and of the order \\(12m - 22\\) in the coordinates \\((x, y, z)\\).\n\n11. But throwing out the factor \\( s^2 \\), and observing that in the first line the quadric functions of \\( m \\) are each a numerical multiple of \\( 51m^2 - 198m + 183 \\), the condition becomes\n\n\\[\n0 = (51m^2 - 198m + 183)H^2(3H\\Phi - 2\\Phi H) \\\\\n+ \\{ (-96m + 168)H(\\partial \\cdot \\nabla)H + (-90m + 162)H\\partial \\nabla H + (120m - 240)\\partial \\nabla H \\} \\\\\n+ \\{ 9H^2\\Omega - 45H\\Omega \\partial H + 40\\Psi \\partial H \\}.\n\\]\n\nArticle Nos 12 & 13.—Second transformation.\n\n12. We effect this by means of the formula\n\n\\[\n(m-2)(3H\\Phi - 2\\Phi H) = -\\mathfrak{J} \\text{ Jac.}(U, \\Phi, H), \\ldots \\ldots (J)*\n\\]\n\n* (J) here and elsewhere refers to the Jacobian Formula, see post, Article Nos. 34 & 35.\nfor substituting this value of \\((3H\\partial\\Phi - 2\\Phi\\partial H)\\) the equation becomes divisible by \\(S\\); and dividing out accordingly, the condition becomes\n\n\\[\n\\frac{51m^2 - 198m + 183}{m-2} H^2 \\text{Jac.}(U, \\Phi, H)\n\\]\n\n\\[\n+ (-96m + 168)H^2(\\partial \\cdot \\nabla)H + (-90m + 162)H^2\\partial \\nabla H + (120m - 240)H\\partial H\\nabla H\n\\]\n\n\\[\n+S(9H^2\\partial \\Omega - 45H\\Omega\\partial H + 40\\Psi\\partial H)=0.\n\\]\n\n13. We have (see post, Article Nos. 36 to 40)\n\n\\[\n\\text{Jac.}(U, \\Phi, H) = -(\\partial \\cdot \\nabla)H;\n\\]\n\nand introducing also \\(\\partial \\cdot \\nabla H\\) in place of \\(\\partial \\nabla H\\) by means of the formula\n\n\\[\n\\partial \\nabla H = \\partial (\\nabla H) - (\\partial \\cdot \\nabla)H,\n\\]\n\nthe condition becomes\n\n\\[\n\\left\\{\\frac{51m^2 - 198m + 183}{m-2} - (6m - 6)\\right\\}H^2(\\partial \\cdot \\nabla)H\n\\]\n\n\\[\n+ (-90m + 162)H^2(\\nabla H) + 120(m - 2)H\\partial H\\nabla H\n\\]\n\n\\[\n+S(9H^2\\partial \\Omega - 45H\\Omega\\partial H + 40\\Psi\\partial H)=0,\n\\]\n\nor, as this may be written,\n\n\\[\n(45m^2 - 180m + 171)H^2(\\partial \\cdot \\nabla)H\n\\]\n\n\\[\n+ (-90m + 162)(m - 2)H^2(\\nabla H) + 120(m - 2)^2H\\partial H\\nabla H\n\\]\n\n\\[\n+(m - 2)S(9H^2\\partial \\Omega - 45H\\Omega\\partial H + 40\\Psi\\partial H)=0.\n\\]\n\nArticle Nos. 14 to 17.—Third transformation.\n\n14. We have the following formulæ,\n\n\\[\nS \\text{Jac.}(U, \\nabla H, H) - (5m - 11)\\partial H\\nabla H + (3m - 6)H\\partial (\\nabla H) = 0, \\ldots \\ldots (J)\n\\]\n\n\\[\nS \\text{Jac.}(U, \\nabla, H)H - (2m - 4)\\partial H\\nabla H + (3m - 6)H(\\partial \\cdot \\nabla)H = 0, \\ldots \\ldots (J)\n\\]\n\nin the latter of which, treating \\(\\nabla\\) as a function of the coordinates, we first form the symbol \\(\\text{Jac.}(U, \\nabla, H)\\), and then operating therewith on \\(H\\), we have \\(\\text{Jac.}(U, \\nabla, H)H\\); these give\n\n\\[\nH\\partial (\\nabla H) = \\frac{5m - 11}{3(m - 2)}\\partial H\\nabla H - \\frac{S}{3(m - 2)}\\text{Jac.}(U, \\nabla H, H),\n\\]\n\n\\[\nH(\\partial \\cdot \\nabla)H = \\frac{S}{3}\\partial H\\nabla H - \\frac{S}{3(m - 2)}\\text{Jac.}(U, \\nabla, H)H;\n\\]\n\nand substituting these values, the resulting coefficient of \\(H\\partial H\\nabla H\\) is\n\n\\[\n(45m^2 - 180m + 171)\\frac{S}{3}\n\\]\n\n\\[\n+ (-90m + 162)\\frac{5m - 11}{3}\n\\]\n\n\\[\n+ 120(m - 2)^2,\n\\]\n\nwhich is \\(= 0\\).\n15. Hence the condition will contain the factor $\\Omega$, and throwing out this, and also the constant factor $\\frac{1}{m-2}$, it becomes\n\n$$(-15m^2 + 60m - 57)H \\text{Jac.}(U, \\nabla, H)H$$\n$$+(30m - 54)(m-2)H \\text{Jac.}(U, \\nabla H, H)$$\n$$+(m-2)^2(9H^2\\Omega - 45H\\Omega\\partial H + 40\\Psi\\partial H)=0.$$\n\n16. We have\n\n$$\\partial_x(\\nabla H)=(\\partial_x \\cdot \\nabla)H + \\partial_x \\nabla H,$$\n\nviz. in $(\\partial_x \\cdot \\nabla)H$, treating $\\nabla$ as a function of $(x, y, z)$ we operate upon it with $\\partial_x$ to obtain the new symbol $\\partial_x \\cdot \\nabla$, and with this we operate on $H$; in $\\partial_x \\nabla$ we simply multiply together the symbols $\\partial_x$ and $\\nabla$, giving a new symbol of the form $(\\partial_x^2, \\partial_y \\partial_x, \\partial_z \\partial_x)$ which then operates on $H$. We have the like values of $\\partial_y(\\nabla H)$ and $\\partial_z(\\nabla H)$; and thence also\n\n$$\\text{Jac.}(U, \\nabla H, H)=\\text{Jac.}(U, \\nabla, H)H + \\text{Jac.}(U, \\nabla H, H),$$\n\nviz. in the determinant $\\text{Jac.}(U, \\nabla, H)$ the second line corresponding to $\\nabla$ is $\\partial_x \\cdot \\nabla$, $\\partial_y \\cdot \\nabla$, $\\partial_z \\cdot \\nabla$ ($\\nabla$ being the operand); and the Jacobian thus obtained is a symbol which operates on $H$ giving $\\text{Jac.}(U, \\nabla, H)H$; and in the determinant $\\text{Jac.}(U, \\nabla H, H)$ the second line is $\\partial_x \\nabla H$, $\\partial_y \\nabla H$, $\\partial_z \\nabla H$ ($\\nabla$ being simply multiplied by $\\partial_x$, $\\partial_y$, $\\partial_z$ respectively).\n\n17. Substituting, the condition becomes\n\n$$(-15m^2 + 60m - 57)H \\text{Jac.}(U, \\nabla, H)H$$\n$$+(30m - 54)(m-2)\\{\\text{H Jac.}(U, \\nabla, H)H + \\text{Jac.}(U, \\nabla H, H)\\}$$\n$$+(m-2)^2\\{9H^2\\Omega - 54H\\Omega\\partial H + 40\\Psi\\partial H\\}=0,$$\n\nor, what is the same thing,\n\n$$(15m^2 - 54m + 51)H \\text{Jac.}(U, \\nabla, H)H$$\n$$+(30m - 54)(m-2)H \\text{Jac.}(U, \\nabla H, H)$$\n$$+(m-2)^2\\{9H^2\\Omega - 45H\\Omega\\partial H + 40\\Psi\\partial H\\}=0.$$\n\nArticle Nos. 18 to 27.—Fourth transformation, and final form of the condition for a Sextactic Point.\n\n18. I write\n\n$$(5m-12)\\Omega\\partial H-(3m-6)H\\partial \\Omega=\\Omega \\text{Jac.}(U, \\Omega, H) \\ldots \\ldots (J)$$\n\n$$\\Omega\\partial H+H\\partial \\Omega=\\partial(\\Omega H),$$\n\nand, introducing for convenience the new symbol 'W,\n\n$$-5\\Omega\\partial H+H\\partial \\Omega=W,$$\n\nso that\n\n$$\\begin{vmatrix}\n5m-12, & -(3m-6), & \\Omega \\text{Jac.}(U, \\Omega, H) \\\\\n1, & 1, & \\partial(\\Omega H) \\\\\n-5, & 1, & W\n\\end{vmatrix}=0.$$\nor what is the same thing,\n\n\\[(8m-18)W + 69 \\text{Jac.}(U, \\Omega, H) + (10m-18)\\partial(\\Omega H) = 0,\\]\n\nwe have\n\n\\[W = H\\partial\\Omega - 5\\Omega\\partial H = \\frac{-3}{4m-9}9 \\text{Jac.}(U, \\Omega, H) - \\frac{5m-9}{4m-9}\\partial(\\Omega H).\\]\n\n19. We have also\n\n\\[(8m-18)\\Psi\\partial H - (3m-6)H\\partial\\Psi - 9 \\text{Jac.}(U, \\Psi, H) = 0, \\ldots \\ldots \\ldots (J)\\]\n\nthat is\n\n\\[\\Psi\\partial H = \\frac{1}{4m-9}9 \\text{Jac.}(U, \\Psi, H) + \\frac{3(m-2)}{4m-9}H\\partial\\Psi,\\]\n\nand thence\n\n\\[9HW + 40\\Psi\\partial H\\]\n\n\\[= 9H\\partial\\Omega - 45H\\Omega\\partial H + 40\\Psi\\partial H\\]\n\n\\[= -\\frac{9(5m-9)}{4m-9}H\\partial(\\Omega H) + \\frac{60(m-2)}{4m-9}H\\partial\\Psi\\]\n\n\\[+ \\frac{9}{4m-9}\\{-27H \\text{Jac.}(U, \\Omega, H) + 40 \\text{Jac.}(U, \\Psi, H)\\}.\\]\n\n20. The condition thus becomes\n\n\\[(15m^2-54m+51)(4m-9)H \\text{Jac.}(U, \\nabla, H)H\\]\n\n\\[+ 6(5m-9)(m-2)(4m-9)H \\text{Jac.}(U, \\nabla H, H)\\]\n\n\\[+ 3(m-2)\\{-3(5m-9)(m-2)H\\partial(\\Omega H) + 20(m-2)^2H\\partial\\Psi\\}\\]\n\n\\[+ (m-2)^2\\{-27H \\text{Jac.}(U, \\Omega, H) + 40 \\text{Jac.}(U, \\Psi, H)\\} = 0,\\]\n\nwhich for shortness I represent by\n\n\\[3HII + (m-2)^2\\{-27H \\text{Jac.}(U, \\Omega, H) + 40 \\text{Jac.}(U, \\Psi, H)\\} = 0,\\]\n\nso that we have\n\n\\[II = (5m^2-18m+17)(4m-9) \\text{Jac.}(U, \\nabla, H)H\\]\n\n\\[+ 2(5m-9)(m-2)(4m-9) \\text{Jac.}(U, \\nabla H, H)\\]\n\n\\[+ (m-2)\\{-3(5m-9)(m-2)\\partial(\\Omega H) + 20(m-2)^2\\partial\\Psi\\}.\\]\n\n21. Write\n\n\\[\\Psi_1 = (\\mathcal{A}', \\mathcal{B}', \\mathcal{C}', \\mathcal{F}', \\mathcal{G}', \\mathcal{H}')\\mathcal{X}(A, B, C),\\]\n\nwhere \\((A, B, C)\\) are as before the first differential coefficients of \\(U\\), and \\((a', b', c', f', g', h')\\) being the second differential coefficients of \\(H\\), \\((\\mathcal{A}', \\mathcal{B}', \\mathcal{C}', \\mathcal{F}', \\mathcal{G}', \\mathcal{H}')\\) are the inverse coefficients, viz., \\(\\mathcal{A}' = b'c' - f'^2\\), &c. We have\n\n\\[-(m-1)^2\\partial\\Psi_1 = (3m-6)(3m-7)\\partial(\\Omega H) - (3m-7)^2\\partial\\Psi \\quad \\text{(see post, Nos. 41 to 46)},\\]\n\nthat is\n\n\\[(3m-6)\\partial(\\Omega H) = (3m-7)\\partial\\Psi - \\frac{(m-1)^2}{3m-7}\\partial\\Psi_1,\\]\nand thence\n\n\\[ II = (5m^2 - 18m + 17)(4m - 9) \\text{Jac.}(U, \\nabla, H)H \\\\\n+ 2(5m - 9)(m - 2)(4m - 9) \\text{Jac.}(U, \\nabla H, H) \\\\\n+(m - 2)\\left\\{(5m^2 - 18m + 17)\\partial \\Psi + \\frac{(m-1)^2(5m-9)}{3m-7}\\partial \\Psi_1\\right\\} = 0. \\]\n\n22. Now\n\n\\[ \\Psi = (\\mathcal{A}, \\mathcal{B}, \\mathcal{C}, \\mathcal{F}, \\mathcal{G}, \\mathcal{H})(\\mathcal{A}', \\mathcal{B}', \\mathcal{C}')^2, \\quad \\Psi_1 = (\\mathcal{A}', \\mathcal{B}', \\mathcal{C}', \\mathcal{F}', \\mathcal{G}', \\mathcal{H})(\\mathcal{A}, \\mathcal{B}, \\mathcal{C})^2, \\]\n\nand writing for shortness\n\n\\[ E\\Psi = (\\partial \\mathcal{A}, \\ldots, \\mathcal{A}', \\mathcal{B}', \\mathcal{C}')^2, \\quad F\\Psi = (\\mathcal{A}, \\ldots, \\mathcal{A}', \\mathcal{B}', \\mathcal{C}' \\partial \\mathcal{A}', \\partial \\mathcal{B}', \\partial \\mathcal{C}'), \\]\n\\[ E\\Psi_1 = (\\partial \\mathcal{A}', \\ldots, \\mathcal{A}, \\mathcal{B}, \\mathcal{C})^2, \\quad F\\Psi_1 = (\\mathcal{A}', \\ldots, \\mathcal{A}, \\mathcal{B}, \\mathcal{C} \\partial \\mathcal{A}, \\partial \\mathcal{B}, \\partial \\mathcal{C}), \\]\n\n(we might, in a notation above explained, write \\( E\\Psi = \\partial \\Psi_{ii}, F\\Psi = \\frac{1}{2}\\partial \\Psi_{ii} \\), and in like manner \\( E\\Psi_1 = \\partial \\Psi_{i \\bar{i}}, F\\Psi_1 = \\frac{1}{2}\\partial \\Psi_{i \\bar{i}} \\)), then we have\n\n\\[ \\partial \\Psi = E\\Psi + 2F\\Psi, \\quad \\partial \\Psi_1 = E\\Psi_1 + 2F\\Psi_1. \\]\n\nWe have moreover\n\n\\[ \\text{Jac.}(U, \\nabla H, H) = -\\frac{m-1}{3m-7}E\\Psi_1, \\quad \\text{post, Nos. 47 to 50.} \\]\n\\[ \\text{Jac.}(U, \\nabla, H)H = -E\\Psi, \\quad \\text{post, Nos. 51 to 53.} \\]\n\n23. The just-mentioned formulæ give\n\n\\[ II = -(5m^2 - 18m + 17)(4m - 9)E\\Psi \\\\\n- 2(5m - 9)(m - 2)(4m - 9)\\frac{m-1}{3m-7}F\\Psi_1 \\\\\n+(m - 2)(5m^2 - 18m + 17)(E\\Psi + 2F\\Psi) \\\\\n+\\frac{(5m - 9)(m - 1)^2(m - 2)}{3m - 7}(E\\Psi_1 + 2F\\Psi_1), \\]\n\nthat is\n\n\\[ II = -(3m - 7)(5m^2 - 18m + 17)E\\Psi \\\\\n+ 2(m - 2)(5m^2 - 18m + 17)F\\Psi \\\\\n+\\frac{(5m - 9)(m - 1)^2(m - 2)}{3m - 7}E\\Psi_1 \\\\\n-\\frac{2(m - 1)(m - 2)(3m - 8)(5m - 9)}{3m - 7}F\\Psi_1, \\]\n\nor, as this may also be written,\n\n\\[ (3m - 7)II = -(5m^2 - 18m + 17)\\{-2(m - 1)(m - 2)F\\Psi_1 \\\\\n-(5m - 9)(m - 2)\\{(m - 1)(3m - 8)F\\Psi_1 + (3m - 7)(3m - 8)F\\Psi - (m - 1)^2E\\Psi_1\\} \\\\\n+(25m^2 - 103m + 106)(m - 2)\\{- (m - 1)F\\Psi_1 + (3m - 7)F\\Psi\\}. \\]\n24. But recollecting that\n\n\\[ \\Omega = (\\mathbf{A}, \\mathbf{B}, \\mathbf{C}, \\mathbf{F}, \\mathbf{G}, \\mathbf{H})(\\partial_x, \\partial_y, \\partial_z)^3 H \\]\n\n\\[ = (\\mathbf{A}, \\mathbf{B}, \\mathbf{C}, \\mathbf{F}, \\mathbf{G}, \\mathbf{H})(a', b', c', 2f', 2g', 2h'), \\]\n\nand putting\n\n\\[ E\\Omega = (\\partial A, \\ldots \\times a', \\ldots) \\quad (= \\partial \\Omega_H), \\]\n\n\\[ F\\Omega = (\\mathbf{A}, \\ldots \\times \\partial a', \\ldots) \\quad (= \\partial \\Omega_U), \\]\n\nwe have, post, Nos. 41 to 46,\n\n\\[ -2(m-1)(m-2) F\\Psi_1 + (3m-7)^2 E\\Psi = (3m-6)(3m-7)HE\\Omega \\]\n\n\\[ (m-1)(3m-8)F\\Psi_1 + (3m-7)(3m-8)F\\Psi - (m-1)^2 E\\Psi_1 = (3m-6)(3m-7)HF\\Omega \\]\n\n\\[ -(m-1)F\\Psi_1 + (3m-7)F\\Psi = (3m-7)\\Omega\\partial H, \\]\n\nand the foregoing equation becomes\n\n\\[ (3m-7)\\Pi = -(5m^2-18m+17)(3m-6) \\quad (3m-7)HE\\Omega \\]\n\n\\[ -(5m-9)(m-2)(3m-6) \\quad (3m-7)HF\\Omega \\]\n\n\\[ +(m-2)(25m^2-103m+106)(3m-7)\\Omega\\partial H. \\]\n\n25. But we have\n\n\\[ \\S Jac.(U, H, \\Omega_H) - (3m-6)HE\\Omega + (2m-4)\\Omega\\partial H = 0, \\ldots \\ldots (J) \\]\n\n\\[ \\S Jac.(U, H, \\Omega_U) - (3m-6)HF\\Omega + (3m-6)\\Omega\\partial H = 0, \\ldots \\ldots (J) \\]\n\nthat is\n\n\\[ 3(m-2)HE\\Omega = 2(m-2)\\Omega\\partial H + \\S Jac.(U, H, \\Omega_H), \\]\n\n\\[ 3(m-2)HF\\Omega = (3m-8)\\Omega\\partial H + \\S Jac.(U, H, \\Omega_U), \\]\n\nand we thus obtain\n\n\\[ \\Pi = -(5m^2-18m+17)\\{2(m-2)\\Omega\\partial H + \\S Jac.(U, H, \\Omega_H)\\} \\]\n\n\\[ -(5m-9)(m-2)\\{(3m-8)\\Omega\\partial H + \\S Jac.(U, H, \\Omega_U)\\} \\]\n\n\\[ +(25m^2-103m+106)(m-2)\\Omega\\partial H, \\]\n\nwhere the coefficient of \\((m-2)\\Omega\\partial H\\) is\n\n\\[ -(10m^2-36m+34) \\]\n\n\\[ -(5m-9)(3m-8) \\]\n\n\\[ +(25m^2-103m+106), \\]\n\nwhich is \\(= 0\\). Hence\n\n\\[ \\Pi = -(5m^2-18m+17)\\S Jac.(U, H, \\Omega_H) \\]\n\n\\[ -(5m-9)(m-2)\\S Jac.(U, H, \\Omega_U). \\]\n\n26. Substituting this in the equation\n\n\\[ 3H\\Pi + (m-2)^2\\{-27H Jac.(U, \\Omega, H) + 40 Jac.(U, \\Psi, H)\\} = 0, \\]\n\nthe result contains the factor \\(\\S\\), and, throwing this out, the condition is\n\n\\[ 3H\\{-5m^2-18m+17\\}Jac.(U, H, \\Omega_H) - (5m-9)(m-2)Jac.(U, H, \\Omega_U) \\]\n\n\\[ +(m-2)^2\\{27H Jac.(U, H, \\Omega) - 40 Jac.(U, H, \\Psi)\\} = 0, \\]\nor, as this may also be written,\n\n\\[-(15m^2 - 54m + 51)H \\text{Jac.}(U, H, \\Omega_H) - 3(5m - 9)(m - 2)H \\text{Jac.}(U, H, \\Omega_U)\\]\n\n\\[+ 27(m - 2)^2 \\{H \\text{Jac.}(U, H, \\Omega_H) + H \\text{Jac.}(U, H, \\Omega_U)\\}\\]\n\n\\[-40(m - 2)^2 \\text{Jac.}(U, H, \\Psi) = 0.\\]\n\n27. Hence the condition finally is\n\n\\[(12m^2 - 54m + 57)H \\text{Jac.}(U, H, \\Omega_H) + (m - 2)(12m - 27)H \\text{Jac.}(U, H, \\Omega_U)\\]\n\n\\[-40(m - 2)^2 \\text{Jac.}(U, H, \\Psi) = 0,\\]\n\nor, as this may also be written,\n\n\\[-3(m - 1)H \\text{Jac.}(U, H, \\Omega_H) + (m - 2)(12m - 27)H \\text{Jac.}(U, H, \\Omega)\\]\n\n\\[-40(m - 2)^2 \\text{Jac.}(U, H, \\Psi) = 0,\\]\n\nviz. the sextactic points are the intersections of the curve \\(m\\) with the curve represented by this equation; and observing that \\(U, H, H\\Omega\\) and \\(\\Psi\\) are of the orders \\(m, 3m - 6, 8m - 18\\) respectively, the order of the curve is as above mentioned \\(= 12m - 27\\).\n\nArticle Nos. 28 to 30.—Application to a Cubic.\n\n28. I have in my former memoir, No. 30, shown that for a cubic curve\n\n\\[\\Omega = (\\mathfrak{A}, \\mathfrak{B}, \\mathfrak{C}, \\mathfrak{F}, \\mathfrak{G}, \\mathfrak{H}, \\mathfrak{O}_x, \\mathfrak{O}_y, \\mathfrak{O}_z)^3H = -2S. U = 0,\\]\n\nthis implies \\(\\text{Jac.}(U, H, \\Omega) = 0\\), and hence if one of the two Jacobians, \\(\\text{Jac.}(U, H, \\Omega_U)\\), \\(\\text{Jac.}(U, H, \\Omega_H)\\) vanish, the other will also vanish. Now, using the canonical form\n\n\\[U = x^3 + y^3 + z^3 + 6xyz,\\]\n\nwe have\n\n\\[\\Omega = (\\mathfrak{A}, \\ldots, \\mathfrak{d}', \\ldots)\\]\n\n\\[(yz - l^2x^2, zx - l^2y^2, xy - l^2z^2, l^2yz - lx^2, l^2zx - ly^2, l^2xy - lz^2)\\]\n\n\\[-3l^2x, -3l^2y, -3l^2z, (1 + 2l^3)x, (1 + 2l^3)y, (1 + 2l^3)z\\),\n\nthe development of which in fact gives the last-mentioned result. But applying this formula to the calculation of \\(\\text{Jac.}(U, H, \\Omega_U)\\), then disregarding numerical factors, we have\n\n\\[\\partial_x \\Omega_U = (yz - l^2x^2, \\ldots, l^2yz - lx^2, \\ldots, -3l^2, 0, 0, (1 + 2l^3), 0, 0)\\]\n\n\\[-3l^2(yz - l^2x^2)\\]\n\n\\[(1 + 2l^3)(l^2yz - lx^2)\\]\n\n\\[= (-l + l^3)(x^2 + 2lyz) = S\\partial_x U;\\]\n\nand in like manner\n\n\\[\\partial_y \\Omega_U = S\\partial_y U, \\quad \\partial_z \\Omega_U = S\\partial_z U,\\]\n\nand therefore\n\n\\[\\text{Jac.}(U, H, \\Omega_U) = S\\text{Jac.}(U, H, U) = 0,\\]\nwhence also\n\n\\[ \\text{Jac.} (U, H, \\Omega_H) = 0; \\]\n\nand the condition for a sextactic point assumes the more simple form,\n\n\\[ \\text{Jac.} (U, H, \\Psi) = 0. \\]\n\n29. Now (former memoir, No. 32) we have\n\n\\[\n\\Psi = (\\mathcal{A}, \\mathcal{B}, \\mathcal{C}, \\mathcal{F}, \\mathcal{G}, \\mathcal{H}) \\partial_x H, \\partial_y H, \\partial_z H)^2 \\\\\n= (1 + 8l^6)(y^3z^3 + z^3x^3 + x^3y^3) \\\\\n+ (-9l^6)(x^3 + y^3 + z^3)^2 \\\\\n+ (-2l - 5l^4 - 20l^6)(x^3 + y^3 + z^3)xyz \\\\\n+ (-15l^2 - 78l^6 + 12l^8)x^3y^3z^3,\n\\]\n\nor observing that \\(x^3 + y^3 + z^3\\) and \\(xyz\\), and therefore the last three lines of the expression of \\(\\Psi\\) are functions of \\(U (= x^3 + y^3 + z^3 + 6lxyz)\\) and \\(H (= -l(x^3 + y^3 + z^3) + (1 + 2l^3)xyz)\\), and consequently give rise to the term \\(= 0\\) in \\(\\text{Jac.} (U, H, \\Psi)\\), we may write\n\n\\[\n\\Psi = (1 + 8l^6)(y^3z^3 + z^3x^3 + x^3y^3).\n\\]\n\n30. We have then, disregarding a constant factor,\n\n\\[\n\\text{Jac.} (U, H, \\Psi) = \\text{Jac.} (x^3 + y^3 + z^3, xyz, y^3z^3 + z^3x^3 + x^3y^3) \\\\\n= \\begin{vmatrix}\nx^3 & y^3 & z^2 \\\\\nyz & zx & xy \\\\\nx^2(y^3 + z^3) & y^2(z^3 + x^3) & z^2(x^3 + y^3)\n\\end{vmatrix} \\\\\n= x^3(y^6 - z^6) + y^3(z^6 - x^6) + z^3(x^6 - y^6) \\\\\n= (y^3 - z^3)(z^3 - x^3)(x^3 - y^3),\n\\]\n\nso that the sextactic points are the intersections of the curve\n\n\\[ U = x^3 + y^3 + z^3 + 6lxyz = 0, \\]\n\nwith the curve\n\n\\[ (y^3 - z^3)(z^3 - x^3)(x^3 - y^3) = 0. \\]\n\nArticle Nos. 31 to 33.—Proof of identities for the first transformation.\n\n31. Calculation of \\((\\partial_1^5 + 10\\partial_1^3\\partial_2 + 10\\partial_1^2\\partial_3 + 15\\partial_1\\partial_2^2)U.\\)\n\nWriting \\(\\partial\\) in place of \\(D\\), we have (former memoir, No. 20)\n\n\\[\n(\\partial_1^5 + 6\\partial_1^3\\partial_2)U = \\frac{s^2}{(m-1)^3}(-2\\partial_2H - \\partial_2^2H + \\frac{3m-6}{m-1}H\\Phi - \\frac{2s}{m-1}\\nabla H).\n\\]\n\nBut\n\n\\[\n-2\\partial_2H = \\frac{6m-12}{m-1}H\\Phi - \\frac{2s}{m-1}\\nabla H, \\\\\n-\\partial_2^2H = \\frac{(3m-6)(3m-7)}{(m-1)^2}H\\Phi - \\frac{6m-14}{(m-1)^2}\\nabla H + \\frac{s^2}{(m-1)}\\Omega, \\quad \\text{former memoir, Nos. 21 & 22;}\n\\]\nand thence\n\n\\[\n(\\partial_1^4 + 6\\partial_1^2\\partial_2)U = \\frac{S^2}{(m-1)^4}(18m^2 - 66m + 60)H\\Phi \\\\\n+ \\frac{S^3}{(m-1)^4}(-10m + 18)\\nabla H \\\\\n+ \\frac{S^4}{(m-1)^4}(\\Omega);\n\\]\n\nwhence operating on each side with \\(\\partial_1 = \\partial\\), we have\n\n\\[\n(\\partial_1^5 + 10\\partial_1^3\\partial_2 + 6\\partial_1^2\\partial_3 + 12\\partial_1\\partial_2^2)U = \\frac{S^2}{(m-1)^4}(18m^2 - 66m + 60)(H\\partial\\Phi + \\Phi\\partial H) \\\\\n+ \\frac{S^3}{(m-1)^4}(-10m + 18)\\{\\partial \\cdot \\nabla\\}H + \\partial \\nabla H \\\\\n+ \\frac{S^4}{(m-1)^4}\\partial \\Omega.\n\\]\n\nWe have besides (see Appendix, Nos. 69 to 74),\n\n\\[\n\\partial_1^2\\partial_3 U = \\frac{S^2}{(m-1)^3}\\{(3m - 6)H\\partial\\Phi + (-m + 3)\\Phi\\partial H\\} \\\\\n+ \\frac{S^3}{(m-1)^3}\\{-\\partial \\cdot \\nabla\\}H\\},\n\\]\n\n\\[\n\\partial_1\\partial_2^2 U = \\frac{S^2}{(m-1)^3}(-H\\partial\\Phi + \\Phi\\partial H);\n\\]\n\nand thence\n\n\\[\n(4\\partial_1^3\\partial_3 + 3\\partial_1\\partial_2^2)U = \\frac{S^2}{(m-1)^3}\\{(90m - 21)H\\partial\\Phi + (-m + 9)\\Phi\\partial H\\} \\\\\n+ \\frac{S^3}{(m-1)^3}\\{-4\\partial \\cdot \\nabla\\}H\\};\n\\]\n\nand adding this to the foregoing expression for\n\n\\[\n(\\partial_1^5 + 10\\partial_1^3\\partial_2 + 6\\partial_1^2\\partial_3 + 12\\partial_1\\partial_2^2)U,\n\\]\n\nwe have\n\n\\[\n(\\partial_1^5 + 10\\partial_1^3\\partial_2 + 10\\partial_1^2\\partial_3 + 15\\partial_1\\partial_2^2)U = \\\\\n\\frac{S^2}{(m-1)^4}\\{(27m^2 - 96m + 81)H\\partial\\Phi + (17m^2 - 56m + 51)\\Phi\\partial H\\} \\\\\n+ \\frac{S^3}{(m-1)^4}\\{-14m + 22\\}\\partial \\cdot \\nabla\\}H + (-10m + 18)\\partial \\nabla \\cdot H\\} \\\\\n+ \\frac{S^4}{(m-1)^4}\\partial \\Omega.\n\\]\n\n32. Calculation of\n\n\\[\n\\partial_4 U\\partial_1 H + 2\\partial_3 U\\partial_2 H + 2\\partial_2 U\\partial_3 H.\n\\]\nWe have\n\n\\[ \\partial_4 U = \\frac{3^2}{(m-1)^2} \\frac{3}{3} \\partial_2 H + \\partial^3 H - \\frac{1}{m-1} H \\Phi - \\frac{3^3}{m-1} \\nabla H, \\]\n\n\\[ \\partial_3 U = \\frac{3^2}{(m-1)^2} \\partial H, \\]\n\n\\[ \\partial_2 U = \\frac{3^2}{(m-1)^2} H, \\]\n\nfor which values see Appendix, No. 58. And hence the expression sought for is\n\n\\[ = \\frac{3^2}{(m-1)^3} \\left[ (m-1)(\\frac{3}{3} \\partial_2 H + \\partial^3 H) - H \\Phi - \\frac{3^3}{3} \\nabla H \\right] \\partial H \\]\n\n\\[ + 2(m-1) \\partial H \\partial_2 H \\]\n\n\\[ + 2H((-3m+6)H \\partial \\Phi - \\Phi \\partial H + \\partial (\\partial . \\nabla) H) \\}, \\]\n\nwhich is\n\n\\[ = \\frac{3^2}{(m-1)^3} \\left\\{ \\frac{3}{3}(m-1) \\partial H \\partial_2 H \\right. \\]\n\n\\[ + (m-1) \\partial H \\partial^3 H \\]\n\n\\[ + (-6m+12)H^2 \\partial \\Phi - 3H \\Phi \\partial H \\}\n\n\\[ + \\frac{3^3}{(m-1)^3} \\{ 2H(\\partial . \\nabla) H - \\frac{3}{3} \\partial H \\nabla H \\}. \\]\n\nBut we have, former memoir, Nos. 21 & 25,\n\n\\[ \\partial_2 H = -\\frac{(3m-6)}{m-1} H \\Phi - \\frac{3}{m-1} \\nabla H, \\]\n\n\\[ \\partial^3 H = -\\frac{(3m-6)(3m-7)}{(m-1)^2} H \\Phi + \\frac{6m-14}{(m-1)^2} \\nabla H - \\frac{3^2}{(m-1)^2} \\Omega, \\]\n\nso that the foregoing expression becomes\n\n\\[ = \\frac{3^2}{(m-1)^3} \\left\\{ -(8m-16)H \\Phi \\partial H + \\frac{3}{3} \\partial H \\nabla H \\right. \\]\n\n\\[ - \\frac{(3m-6)(3m-7)}{m-1} H \\Phi \\partial H + \\frac{6m-14}{m-1} \\nabla H \\nabla H - \\frac{3^2}{m-1} \\Omega \\partial H \\]\n\n\\[ - 3H \\Phi \\partial H - (6m-12)H^2 \\partial \\Phi \\}\n\n\\[ + \\frac{3^3}{(m-1)^3} \\{ 2H(\\partial . \\nabla) H - \\frac{3}{3} \\partial H \\nabla H \\}; \\]\n\nor finally\n\n\\[ \\partial_4 U \\partial_1 H + 2\\partial_3 U \\partial_2 H + 2\\partial_2 U \\partial_3 H \\]\n\n\\[ = \\frac{3^2}{(m-1)^4} \\left\\{ (-6m^2+18m-12)H^2 \\partial \\Phi + (-17m^2+60m-55)H \\Phi \\partial H \\right\\} \\]\n\n\\[ + \\frac{3^3}{(m-1)^4} \\{ (2m-2)H(\\partial . \\nabla) H + (8m-16) \\partial H \\nabla H \\} \\]\n\n\\[ + \\frac{3^4}{(m-1)^4} \\{ -\\Omega \\partial H \\}. \\]\n33. Calculation of $\\partial_2 U \\partial_3 U$.\n\nThis is\n\n$$\\frac{\\partial^4}{(m-1)^4} H \\partial H.$$  \n\nArticle Nos. 34 & 35.—The Jacobian Formula.\n\n34. In general, if $P$, $Q$, $R$, $S$ be functions of the degrees $p$, $q$, $r$, $s$ respectively, we have identically\n\n$$\\begin{vmatrix}\npP, & qQ, & rR, & sS \\\\\n\\partial_x P, & \\partial_x Q, & \\partial_x R, & \\partial_x S \\\\\n\\partial_y P, & \\partial_y Q, & \\partial_y R, & \\partial_y S \\\\\n\\partial_z P, & \\partial_z Q, & \\partial_z R, & \\partial_z S \\\\\n\\end{vmatrix} = 0,$$\n\nor, what is the same thing,\n\n$$pP \\text{Jac.}(Q, R, S) - qQ \\text{Jac.}(R, S, P) + rR \\text{Jac.}(S, P, Q) - sS \\text{Jac.}(P, Q, R) = 0.$$\n\nHence in particular if $P = U$, and assuming $U = 0$, we have\n\n$$-qQ \\text{Jac.}(R, S, U) + rR \\text{Jac.}(S, U, Q) - sS \\text{Jac.}(U, Q, R) = 0.$$\n\nIf moreover $Q = S$, and therefore $q = 1$, we have\n\n$$-S \\text{Jac.}(R, S, U) + rR \\text{Jac.}(S, U, S) - sS \\text{Jac.}(U, S, R) = 0;$$\n\nor, as this may also be written,\n\n$$-\\Omega \\text{Jac.}(U, R, S) + rR \\text{Jac.}(U, S, S) - sS \\text{Jac.}(U, S, R) = 0;$$\n\nthat is\n\n$$-\\Omega \\text{Jac.}(U, R, S) + rR \\partial S - sS \\partial R = 0.$$\n\n35. Particular cases are\n\n$$(2m - 4) \\Phi \\partial H - (3m - 6) H \\partial \\Phi = \\Omega \\text{Jac.}(U, \\Phi, H), \\text{ante}, \\text{No. 12},$$\n\n$$(5m - 11) \\nabla H \\partial H - (3m - 6) H \\partial (\\nabla H) = \\Omega \\text{Jac.}(U, \\nabla H, H), \\text{,, 14},$$\n\n$$(2m - 4) \\nabla : \\partial H - (3m - 6) H \\partial . \\nabla = \\Omega \\text{Jac.}(U, \\nabla , H), \\text{,, ,,},$$\n\n$$(5m - 12) \\Omega \\partial H - (3m - 6) H \\partial \\Omega = \\Omega \\text{Jac.}(U, \\Omega , H), \\text{,, 18},$$\n\n$$(8m - 18) \\Psi \\partial H - (3m - 6) H \\partial \\Psi = \\Omega \\text{Jac.}(U, \\Psi , H), \\text{,, 19},$$\n\n$$(2m - 4) \\Omega \\partial H - (3m - 6) HE \\Omega = \\Omega \\text{Jac.}(U, \\Omega_H , H), \\text{,, 25},$$\n\n$$(3m - 8) \\Omega \\partial H - (3m - 6) HF \\Omega = \\Omega \\text{Jac.}(U, \\Omega_{\\bar{H}} , H), \\text{,, ,,},$$\n\nwhere it is to be observed that in the third of these formulæ I have, in accordance with the notation before employed, written $\\partial . \\nabla$ to denote the result of the operation $\\partial$ performed on $\\nabla$ as operand. I have also written $\\nabla : \\partial H$ to show that the operation $\\nabla$ is not to be performed on the following $\\partial H$ as an operand, but that it remains as an unperformed operation. As regards the last two equations, it is to be remarked that the demonstration in the last preceding number depends merely on the homogeneity of the functions, and the orders of these functions: in the former of the two formulæ, the\ndifferentiation of $\\Omega$ is performed upon $\\Omega$ in regard to the coordinates $(x, y, z)$ in so far only as they enter through $U$, and $\\Omega$ is therefore to be regarded as a function of the order $2m-4$; in the latter of the two formulæ the differentiation is to be performed in regard to the coordinates in so far only as they enter through $H$, and $\\Omega$ is therefore to be regarded as a function of the order $3m-8$. The two formulæ might also be written\n\n\\[\n(2m-4)\\Omega \\partial H - (3m-6)H \\partial \\Omega_H = \\Omega \\text{ Jac.} (U, \\Omega_H, H),\n\\]\n\n\\[\n(3m-8)\\Omega \\partial H - (3m-6)H \\partial \\Omega_U = \\Omega \\text{ Jac.} (U, \\Omega_U, H);\n\\]\n\nand it may be noticed that, adding these together, we obtain the foregoing formula,\n\n\\[\n(5m-12)\\Omega \\partial H - (3m-6)H \\partial \\Omega = \\Omega \\text{ Jac.} (U, \\Omega, H).\n\\]\n\nArticle Nos. 36 to 40.—Proof of equation $(\\partial \\cdot \\nabla)H = \\text{Jac.}(U, H, \\Phi)$, used in the second transformation.\n\n36. We have\n\n\\[\n\\nabla = (\\lambda, \\mu, \\nu)(\\partial_x, \\partial_y, \\partial_z)\n\\]\n\n\\[\n= (\\lambda \\partial_x + \\mu \\partial_y + \\nu \\partial_z, \\lambda \\partial_x + \\mu \\partial_y + \\nu \\partial_z, \\lambda \\partial_x + \\mu \\partial_y + \\nu \\partial_z).\n\\]\n\nAlso\n\n\\[\n\\partial = (Bv - C\\mu) \\partial_z + (C\\lambda - Av) \\partial_y + (A\\mu - B\\lambda) \\partial_z\n\\]\n\n\\[\n= \\lambda P + \\mu Q + \\nu R,\n\\]\n\nif for a moment\n\n\\[\nP, Q, R = C\\partial_y - B\\partial_z, A\\partial_x - C\\partial_z, B\\partial_x - A\\partial_y.\n\\]\n\nHence\n\n\\[\n\\partial \\cdot \\nabla = (P\\lambda + Q\\mu + Rv).(\\lambda \\partial_x + \\mu \\partial_y + \\nu \\partial_z, \\lambda \\partial_x + \\mu \\partial_y + \\nu \\partial_z, \\lambda \\partial_x + \\mu \\partial_y + \\nu \\partial_z),\n\\]\n\nviz. coefficient of $\\lambda^2$\n\n\\[\n= P\\lambda \\partial_x + P\\mu \\partial_y + P\\nu \\partial_z,\n\\]\n\nand so for the other terms; whence also in $(\\partial \\cdot \\nabla)H$ the coefficients of $\\lambda^2$, &c. are\n\n\\[\n(P\\lambda \\partial_x + P\\mu \\partial_y + P\\nu \\partial_z)H, \\text{ &c.}\n\\]\n\n37. Again, in Jac. $(U, H, \\Phi)$, where $\\Phi = (\\lambda, \\mu, \\nu)^3$, the coefficients of $\\lambda^2$, &c. are Jac. $(U, H, \\lambda)$, &c.; and hence the assumed equation\n\n\\[\n(\\partial \\cdot \\nabla)H = \\text{Jac.} (U, H, \\Phi),\n\\]\n\nin regard to the term in $\\lambda^2$, is\n\n\\[\n(P\\lambda \\partial_x + P\\mu \\partial_y + P\\nu \\partial_z)H = \\text{Jac.} (U, H, \\lambda),\n\\]\n\nand we have\n\n\\[\n\\text{Jac.} (U, H, \\lambda) = \\begin{vmatrix}\nA, B, C \\\\\n\\partial_x H, \\partial_y H, \\partial_z H \\\\\n\\partial_x, \\partial_y, \\partial_z\n\\end{vmatrix}\n\\]\n\n\\[\n= [\\partial_x H(C\\partial_y - B\\partial_z) + \\partial_y H(A\\partial_x - C\\partial_z) + \\partial_z H(B\\partial_x - A\\partial_y)]\\lambda\n\\]\n\n\\[\n= (\\partial_x H.P + \\partial_y H.Q + \\partial_z H.R)\\lambda;\n\\]\nso that the equation is\n\n\\[ P \\partial_x H + P \\partial_y H + P \\partial_z H = P \\partial_x H + Q \\partial_y H + R \\partial_z H, \\]\n\nor, as this may be written,\n\n\\[ [(B \\partial_x - C \\partial_y) \\mathcal{H} - (C \\partial_x - A \\partial_z) \\mathcal{A}] \\partial_y H + [(B \\partial_x - C \\partial_y) \\mathcal{G} - (A \\partial_y - B \\partial_x) \\mathcal{A}] \\partial_x H = 0. \\]\n\n38. The coefficient of \\( \\partial_y H \\) is\n\n\\[ = A \\partial_x \\mathcal{A} + B \\partial_y \\mathcal{H} - C (\\partial_x \\mathcal{A} + \\partial_y \\mathcal{H}), \\]\n\nwhich, in virtue of the identity, post, No. 40,\n\n\\[ \\partial_x \\mathcal{A} + \\partial_y \\mathcal{H} + \\partial_z \\mathcal{G} = 0, \\]\n\nis\n\n\\[ = A \\partial_x \\mathcal{A} + B \\partial_y \\mathcal{H} + C \\partial_z \\mathcal{G}. \\]\n\nAnd in like manner the coefficient of \\( \\partial_z H \\)\n\n\\[ = -(A \\partial_y \\mathcal{A} + B \\partial_y \\mathcal{H} + C \\partial_y \\mathcal{G}), \\]\n\nso that the equation is\n\n\\[ (A \\partial_x \\mathcal{A} + B \\partial_y \\mathcal{H} + C \\partial_z \\mathcal{G}) \\partial_y H - (A \\partial_y \\mathcal{A} + B \\partial_y \\mathcal{H} + C \\partial_y \\mathcal{G}) \\partial_z H = 0. \\]\n\n39. But we have\n\n\\[ \\mathcal{A}a + \\mathcal{H}h + \\mathcal{G}g = H, \\]\n\\[ \\mathcal{A}h + \\mathcal{H}b + \\mathcal{G}f = 0, \\]\n\\[ \\mathcal{A}g + \\mathcal{H}f + \\mathcal{G}c = 0, \\]\n\nor multiplying by \\( x, y, z \\) and adding,\n\n\\[ (m-1)(\\mathcal{A}A + \\mathcal{H}B + \\mathcal{G}C) = xH; \\]\n\nwhence also\n\n\\[ (m-1)(\\mathcal{A}h + \\mathcal{H}b + \\mathcal{G}c + A \\partial_y \\mathcal{A} + B \\partial_y \\mathcal{H} + C \\partial_y \\mathcal{G}) = x \\partial_y H, \\]\n\nthat is\n\n\\[ (m-1)(A \\partial_y \\mathcal{A} + B \\partial_y \\mathcal{H} + C \\partial_y \\mathcal{G}) = x \\partial_y H; \\]\n\nand in like manner\n\n\\[ (m-1)(A \\partial_z \\mathcal{A} + B \\partial_z \\mathcal{H} + C \\partial_z \\mathcal{G}) = x \\partial_z H, \\]\n\nwhence the equation in question. The terms in \\( \\lambda^2 \\) are thus shown to be equal, and it might in a similar manner be shown that the terms in \\( \\mu \\nu \\) are equal; the other terms will then be equal, and we have therefore\n\n\\[ (\\partial \\cdot \\nabla) H = \\text{Jac.} (U, H, \\Phi). \\]\n\n40. The identity\n\n\\[ \\partial_x \\mathcal{A} + \\partial_y \\mathcal{H} + \\partial_z \\mathcal{G} = 0 \\]\n\nassumed in the course of the foregoing proof is easily proved. We have in fact\n\n\\[ \\partial_x \\mathcal{A} + \\partial_y \\mathcal{H} + \\partial_z \\mathcal{G} = \\partial_x (bc - f^2) + \\partial_y (fy - ch) + \\partial_z (fh - bg) \\]\n\\[ = b(\\partial_x c - \\partial_y g) + c(\\partial_x b - \\partial_y h) + f(-2\\partial_x f + \\partial_y g + \\partial_z h) + g(\\partial_x f - \\partial_z b) + h(-\\partial_y c + \\partial_z f), \\]\nwhere the coefficients of \\( b, c, f, g, h \\) separately vanish: we have of course the system\n\n\\[\n\\begin{align*}\n\\partial_x A + \\partial_y B + \\partial_z G &= 0, \\\\\n\\partial_x B + \\partial_y C + \\partial_z F &= 0, \\\\\n\\partial_x G + \\partial_y F + \\partial_z E &= 0.\n\\end{align*}\n\\]\n\nArticle Nos. 41 to 46.—Proof of identities for the fourth transformation.\n\n41. Consider the coefficients \\((a, b, c, f, g, h)\\) and the inverse set \\((A, B, C, F, G, H)\\), and the coefficients \\((a', b', c', f', g', h')\\), and the inverse set \\((A', B', C', F', G', H')\\); then we have identically\n\n\\[\n(a, \\ldots x, y, z)^2(A, \\ldots a, \\ldots) - (A', \\ldots ax + hy + gz, \\ldots)^2 = (a', \\ldots x, y, z)^2(A, \\ldots a', \\ldots) - (A, \\ldots a'x + h'y + g'z, \\ldots)^2,\n\\]\n\nwhere \\((A, \\ldots a, \\ldots)\\) and \\((A, \\ldots a', \\ldots)\\) stand for\n\n\\((A, B, C, F, G, H)(a, b, c, 2f, 2g, 2h)\\)\n\nand\n\n\\((A, B, C, F, G, H)(a', b', c', 2f', 2g', 2h')\\)\n\nrespectively.\n\n42. Taking \\((a, b, c, f, g, h)\\), the second differential coefficients of a function \\(U\\) of the order \\(m\\), and in like manner \\((a', b', c', f', g', h')\\), the second differential coefficients of a function \\(U'\\) of the order \\(m'\\), we have\n\n\\[\nm(m-1)U \\cdot (\\partial_x, \\partial_y, \\partial_z)^2U' - (m-1)^2(A, \\ldots \\partial_x U, \\partial_y U, \\partial_z U)^2 = m'(m'-1)U' \\cdot (\\partial_x, \\partial_y, \\partial_z)^2U - (m'-1)^2(A, \\ldots \\partial_x U', \\partial_y U', \\partial_z U')^2;\n\\]\n\nand in particular if \\(U'\\) be the Hessian of \\(U\\), then \\(m' = 3m - 6\\).\n\n43. Hence writing\n\n\\[\n\\Omega = (\\partial_x, \\partial_y, \\partial_z)^2H, \\quad \\Psi = (\\partial_x, \\partial_y, \\partial_z)^2H, \\quad \\Omega_1 = (\\partial_x, \\partial_y, \\partial_z)^2U, \\quad \\Psi_1 = (\\partial_x, \\partial_y, \\partial_z)^2U,\n\\]\n\nwe have\n\n\\[\nm(m-1)U\\Omega_1 - (m-1)^2\\Psi_1 = (3m-6)(3m-7)H\\Omega - (3m-7)^2\\Psi;\n\\]\n\nor if \\(U = 0\\), then\n\n\\[-(m-1)^2\\Psi_1 = (3m-6)(3m-7)H\\Omega - (3m-7)^2\\Psi;\\]\n\nwhence also\n\n\\[-(m-1)^2\\Psi_1 = (3m-6)(3m-7)(H\\Omega + \\Omega_1) - (3m-7)^2\\Psi,\\]\n\nwhich is the formula, ante No. 21.\n\n44. Recurring to the original formula, since this is an actual identity, we may operate on it with the differential symbol \\(\\partial\\) on the three assumptions,—\n1. \\((a, b, c, f, g, h), (A, B, C, F, G, H)\\) are alone variable.\n2. \\((a', b', c', f', g', h'), (A', B', C', F', G', H')\\) are alone variable.\n3. \\((x, y, z)\\) are alone variable.\nWe thus obtain\n\n\\[\n(\\partial a, \\ldots)(x, y, z)^2(A', \\ldots)(a, \\ldots) = (\\partial a', \\ldots)(x, y, z)^2(A, \\ldots)(a', \\ldots)\n\\]\n\n\\[\n+(a, \\ldots)(x, y, z)^2(A', \\ldots)(a, \\ldots) - (\\partial A, \\ldots)(a'x + hy + gz, \\ldots)^2,\n\\]\n\n\\[\n-2(A', \\ldots)(ax + hy + gz, \\ldots)(\\partial a + y\\partial b + z\\partial c, \\ldots)\n\\]\n\n\\[\n(a, \\ldots)(x, y, z)^2(A', \\ldots)(a, \\ldots) = (\\partial a', \\ldots)(x, y, z)^2(A, \\ldots)(a', \\ldots)\n\\]\n\n\\[\n+(\\partial a', \\ldots)(x, y, z)^2(A, \\ldots)(a', \\ldots)\n\\]\n\n\\[\n-2(A, \\ldots)(a'x + hy + gz, \\ldots)(\\partial a' + y\\partial b' + z\\partial g', \\ldots),\n\\]\n\n\\[\n2(a, \\ldots)(x, y, z)(\\partial x, \\partial y, \\partial z)(A', \\ldots)(a, \\ldots) = 2(a', \\ldots)(x, y, z)(\\partial x, \\partial y, \\partial z)(A, \\ldots)(a', \\ldots)\n\\]\n\n\\[\n-2(A', \\ldots)(ax + hy + gz, \\ldots)(\\partial ax + h\\partial y + g\\partial z, \\ldots)\n\\]\n\n\\[\n-2(A, \\ldots)(a'x + hy + gz, \\ldots)(\\partial a'x + h\\partial y + g\\partial z, \\ldots).\n\\]\n\n45. If in these equations respectively we suppose as before that \\((a, b, c, f, g, h)\\) are the second differential coefficients of a function \\(U\\) of the order \\(m\\), and \\((a', b', c', f', g', h')\\) the second differential coefficients of a function \\(U'\\) of the order \\(m'\\); and that \\((A, B, C)\\), \\((A', B', C')\\) are the first differential coefficients of these functions respectively, then after some easy reductions we have\n\n\\[\n(m-1)(m-2)\\partial U(A', \\ldots)(a, \\ldots) = m'(m'-1)U'(A, \\ldots)(a', \\ldots)\n\\]\n\n\\[\n+m(m-1)U(A', \\ldots)(a, \\ldots) - (m-1)^2(A', \\ldots)(A', B', C')^2,\n\\]\n\n\\[\n-2(m-1)(m-2)(A, \\ldots)(A, B, C)(\\partial A, \\partial B, \\partial C)\n\\]\n\n\\[\nm(m-1)U(A', \\ldots)(a', \\ldots) = (m-1)(m-2)\\partial U'(A, \\ldots)(a', \\ldots)\n\\]\n\n\\[\n+m'(m'-1)U'(A, \\ldots)(a', \\ldots)\n\\]\n\n\\[\n-(m-1)^2(A', \\ldots)(A', B', C')(A', B', C')\n\\]\n\n\\[\n2(m-1)\\partial U(A', \\ldots)(a, \\ldots) = 2(m-1)\\partial U'(A, \\ldots)(a', \\ldots)\n\\]\n\n\\[\n-2(m-1)(A', \\ldots)(A, B, C)(\\partial A, \\partial B, \\partial C)\n\\]\n\n\\[\n-2(m-1)(A', \\ldots)(A', B', C')(A', B', C'),\n\\]\n\nequations which may be verified by remarking that their sum is\n\n\\[\nm(m-1)(\\partial U(A', \\ldots)(a, \\ldots) + U[(A', \\ldots)(a, \\ldots) + (\\partial A', \\ldots)(a, \\ldots)])\n\\]\n\n\\[\n-(m-1)^2(A', \\ldots)(A, B, C)^2 + (A', \\ldots)(A, B, C)(\\partial A, \\partial B, \\partial C) = m'(m'-1) &c.,\n\\]\n\nviz., this is the derivative with \\(\\partial\\) of the equation\n\n\\[\nm(m-1)U(A', \\ldots)(a, \\ldots) - (m-1)^2(A', \\ldots)(A, B, C)^2 = m'(m'-1) &c.\n\\]\n\n46. Taking now \\(U' = H\\), and therefore \\(m' = 3m - 6\\); putting also \\(U = 0, \\partial U = 0\\), and writing as before\n\n\\[\nE\\Psi = (\\partial A, \\ldots)(A', B', C')^2,\n\\]\n\n\\[\nF\\Psi = (A, \\ldots)(A', B', C')(A', B', C'),\n\\]\n\n\\[\nE\\Psi_1 = (\\partial A', \\ldots)(A, B, C)^2,\n\\]\n\n\\[\nF\\Psi_1 = (A', \\ldots)(A, B, C)(\\partial A, \\partial B, \\partial C),\n\\]\n\n\\[\nE\\Omega = (\\partial A, \\ldots)(a', \\ldots),\n\\]\n\n\\[\nF\\Omega = (A, \\ldots)(a', \\ldots),\n\\]\nthen the three equations are\n\n\\[-2(m-1)(m-2)F\\Psi_1 = (3m-6)(3m-7)HE\\Omega - (3m-7)^2E\\Psi,\\]\n\n\\[-(m-1)^2E\\Psi = (3m-7)(3m-8)\\Omega\\partial H + (3m-6)(3m-7)HF\\Omega - 2(3m-7)(3m-8)F\\Psi,\\]\n\n\\[-2(m-1)F\\Psi_1 = 2(3m-7)\\Omega\\partial H - 2(3m-7)F\\Psi,\\]\n\nwhence, adding, we have\n\n\\[-(m-1)^2(E\\Psi_1 + 2F\\Psi_1) = -(3m-7)^2(E\\Psi + 2F\\Psi)\\]\n\n\\[(3m-6)(3m-7)\\{\\Omega\\partial H + H(E\\Omega + F\\Omega)\\}\\]\n\n(that is\n\n\\[-(m-1)^2\\Psi_1 = -(3m-7)^2\\Psi + (3m-6)(3m-7)\\partial .\\Omega H,\\]\n\nwhich is right).\n\nAnd by linearly combining the three equations, we deduce\n\n\\[(3m-6)(3m-7)HE\\Omega = -2(m-1)(m-2)F\\Psi_1 + (3m-7)^2E\\Psi,\\]\n\n\\[(3m-7)\\Omega\\partial H = -(m-1)F\\Psi_1 + (3m-7)F\\Psi,\\]\n\n\\[(3m-6)(3m-7)HF\\Omega = (m-1)(3m-8)F\\Psi_1 + (3m-7)(3m-8)F\\Psi - (m-1)^2E\\Psi_1,\\]\n\nwhich are the formulæ, ante, No. 24.\n\nArticle Nos. 47 to 50.—Proof of an identity used in the fourth transformation, viz.,\n\n\\[Jac.(U, \\nabla H, H) = \\frac{m-1}{3m-7}F\\Psi_1,\\]\n\nor say\n\n\\[Jac.(U, H, \\nabla H) = \\frac{m-1}{3m-7}(A, \\ldots A, B, C)\\partial A, \\partial B, \\partial C).\\]\n\n47. We have\n\n\\[\\nabla = (\\lambda, \\mu, \\nu, \\partial_x, \\partial_y, \\partial_z)\\]\n\n\\[= ((A, B, G), (\\lambda, \\mu, \\nu), (H, B, F), (\\lambda, \\mu, \\nu))\\partial_x, \\partial_y, \\partial_z);\\]\n\nor, attending to the effect of the bar as denoting the exemption of the (A, ...) from differentiation,\n\n\\[Jac.(U, H, \\nabla H) = (A, B, G)\\lambda, \\mu, \\nu) Jac.(U, H, \\partial_x H)\\]\n\n\\[+ (H, B, F)\\lambda, \\mu, \\nu) Jac.(U, H, \\partial_y H)\\]\n\n\\[+ (G, F, E)\\lambda, \\mu, \\nu) Jac.(U, H, \\partial_z H).\\]\n\n48. Now\n\n\\[Jac.(U, H, \\partial_x H) = \\frac{1}{3m-6}Jac.(U, x\\partial_x H + y\\partial_y H + z\\partial_z H, \\partial_x H),\\]\n\nand the last-mentioned Jacobian is\n\n\\[\\partial_x H Jac.(U, x, \\partial_x H) + \\partial_y H Jac.(U, y, \\partial_x H) + \\partial_z H Jac.(U, z, \\partial_x H)\\]\n\n\\[+ y Jac.(U, \\partial_y H, \\partial_x H) + z Jac.(U, \\partial_z H, \\partial_x H),\\]\nwhere the second line is\n\n\\[ = -y \\text{Jac.}(U, \\partial_x H, \\partial_y H) + z \\text{Jac.}(U, \\partial_z H, \\partial_x H), \\]\n\nor writing \\((A', B', C')\\) for the first differential coefficients and \\((a', b', c', f', g', h')\\) for the second differential coefficients of \\(H\\), this is\n\n\\[ = -y \\begin{vmatrix} A, & B, & C \\\\ a', & h', & g' \\\\ h', & b', & f' \\end{vmatrix} + z \\begin{vmatrix} A, & B, & C \\\\ g', & f', & c' \\\\ a', & h', & g' \\end{vmatrix} \\]\n\n\\[ = -y(G', F', C')(A, B, C) + z(H', B', F')(A, B, C). \\]\n\nThe first line is\n\n\\[ = \\begin{vmatrix} A, & B, & C \\\\ A', & B', & C' \\\\ a', & h', & g' \\end{vmatrix} \\]\n\n\\[ = A(B'g' - C'h') + B(C'a' - A'g') + C(A'h' - B'a'), \\]\n\nor reducing by the formulae,\n\n\\[ (3m-7)(A', B', C') = (a'x + h'y + g'z, h'x + b'y + f'z, g'x + f'y + c'z), \\]\n\nthis is\n\n\\[ = \\frac{1}{3m-7} \\left\\{ A(-G'y + H'z) + B(-F'y + B'z) + C(-C'y + F'z) \\right\\} \\]\n\n\\[ = \\frac{1}{3m-7} \\left\\{ -y(G', F', C')(A, B, C) + z(H', B', F')(A, B, C) \\right\\}. \\]\n\nHence we have\n\n\\[ \\text{Jac.}(U, H, \\partial_x H) = \\frac{1}{3m-6} \\left( 1 + \\frac{1}{3m-7} \\right) \\left\\{ -y(G', F', C')(A, B, C) + z(H', B', F')(A, B, C) \\right\\} \\]\n\n\\[ = \\frac{1}{3m-7} \\left\\{ -y(G', F', C')(A, B, C) + z(H', B', F')(A, B, C) \\right\\}; \\]\n\nand in like manner\n\n\\[ \\text{Jac.}(U, H, \\partial_y H) = \\frac{1}{3m-7} \\left\\{ -z(A', H', G')(A, B, C) + x(G', F', C')(A, B, C) \\right\\}, \\]\n\n\\[ \\text{Jac.}(U, H, \\partial_z H) = \\frac{1}{3m-7} \\left\\{ -x(H', B', F')(A, B, C) + y(A', H', G')(A, B, C) \\right\\}. \\]\n\n49. And we thence have\n\n\\[ \\text{Jac.}(U, H, \\nabla H) = \\frac{1}{3m-7} \\begin{vmatrix} (\\bar{A}, \\bar{H}, \\bar{G})(\\lambda, \\mu, \\nu), (\\bar{H}, \\bar{B}, \\bar{F})(\\lambda, \\mu, \\nu), (\\bar{G}, \\bar{F}, \\bar{C})(\\lambda, \\mu, \\nu) \\\\ (\\bar{A}, \\bar{H}, \\bar{G})(A, B, C), (\\bar{H}, \\bar{B}, \\bar{F})(A, B, C), (\\bar{G}, \\bar{F}, \\bar{C})(A, B, C) \\end{vmatrix}_{x, y, z}, \\]\n\nor multiplying the two sides by\n\n\\[ H, = \\begin{vmatrix} a, & h, & g \\\\ h, & b, & f \\\\ g, & f, & c \\end{vmatrix}, \\]\nthe right hand side is\n\n\\[ \\frac{1}{3m-7} \\begin{vmatrix}\nH_\\lambda & H_\\mu & H_\\nu \\\\\nX & Y & Z \\\\\n(m-1)A & (m-1)B & (m-1)C\n\\end{vmatrix} \\]\n\nwhich is\n\n\\[ = H_{\\frac{m-1}{3m-7}} \\begin{vmatrix}\n\\lambda & \\mu & \\nu \\\\\nX & Y & Z \\\\\nA & B & C\n\\end{vmatrix} \\]\n\nif for a moment\n\n\\[ X = (\\mathcal{A}, \\ldots \\mathcal{X} A, B, C)(a, h, g), \\]\n\\[ Y = (\\mathcal{A}, \\ldots \\mathcal{X} A, B, C)(h, b, f), \\]\n\\[ Z = (\\mathcal{A}, \\ldots \\mathcal{X} A, B, C)(g, f, c). \\]\n\n50. Hence observing that these equations may be written\n\n\\[ X = (\\mathcal{A}, \\ldots \\mathcal{X} A, B, C)(\\partial_x A, \\partial_x B, \\partial_x C), \\]\n\\[ Y = (\\mathcal{A}, \\ldots \\mathcal{X} A, B, C)(\\partial_y A, \\partial_y B, \\partial_y C), \\]\n\\[ Z = (\\mathcal{A}, \\ldots \\mathcal{X} A, B, C)(\\partial_z A, \\partial_z B, \\partial_z C), \\]\n\nand that we have\n\n\\[ \\partial = \\begin{vmatrix}\n\\lambda & \\mu & \\nu \\\\\n\\partial_x & \\partial_y & \\partial_z \\\\\nA & B & C\n\\end{vmatrix} \\]\n\nwe obtain for \\( H \\text{ Jac. } (U, H, \\nabla, H) \\) the value\n\n\\[ = H_{\\frac{m-1}{3m-7}}(\\mathcal{A}, \\ldots \\mathcal{X} A, B, C)(\\partial A, \\partial B, \\partial C), \\]\n\nor throwing out the factor \\( H \\), we have the required result.\n\nArticle Nos. 51 to 53.—Proof of identity used in the fourth transformation, viz.,\n\n\\[ \\text{Jac. } (U, \\nabla, H)H = -E\\Psi, \\]\n\nor say\n\n\\[ \\text{Jac. } (U, H, \\nabla)H = (\\partial \\mathcal{A}, \\ldots \\mathcal{X} A', B', C')^2. \\]\n\n51. We have\n\n\\[ \\nabla = ((\\mathcal{A}, \\mathcal{B}, \\mathcal{G})(\\lambda, \\mu, \\nu), (\\mathcal{B}, \\mathcal{F}, \\mathcal{C})(\\lambda, \\mu, \\nu))(\\partial_x, \\partial_y, \\partial_z), \\]\n\nand thence\n\n\\[ \\partial \\cdot \\nabla = ((\\partial_x \\mathcal{A}, \\partial_x \\mathcal{B}, \\partial_x \\mathcal{G})(\\lambda, \\mu, \\nu), (\\partial_x \\mathcal{B}, \\partial_x \\mathcal{F}, \\partial_x \\mathcal{C})(\\lambda, \\mu, \\nu))(\\partial_x, \\partial_y, \\partial_z), \\]\n\nand\n\n\\[ (\\partial_x \\cdot \\nabla)H = ((\\partial_x \\mathcal{A}, \\partial_x \\mathcal{B}, \\partial_x \\mathcal{G})(\\lambda, \\mu, \\nu), (\\partial_x \\mathcal{B}, \\partial_x \\mathcal{F}, \\partial_x \\mathcal{C})(\\lambda, \\mu, \\nu))(\\mathcal{A}', B', C'), \\]\nwith the like values for \\((\\partial_y \\cdot \\nabla)H\\) and \\((\\partial_x \\cdot \\nabla)H\\). And then\n\n\\[\n\\text{Jac.}(U, H, \\nabla)H = \\begin{vmatrix}\nA & B & C \\\\\nA' & B' & C' \\\\\n(\\partial_x \\cdot \\nabla)H & (\\partial_y \\cdot \\nabla)H & (\\partial_z \\cdot \\nabla)H\n\\end{vmatrix},\n\\]\n\nin which the coefficient of \\(A'\\) is\n\n\\[(C\\partial_y - B\\partial_z)(A, B, G)(\\lambda, \\mu, \\nu);\\]\n\nor putting for shortness\n\n\\[(C\\partial_y - B\\partial_z, A\\partial_z - C\\partial_x, B\\partial_x - A\\partial_y) = (P, Q, R);\\]\n\nthe coefficient is\n\n\\[(PA, PB, PG)(\\lambda, \\mu, \\nu).\\]\n\n52. We have\n\n\\[\\partial = (P\\lambda + Q\\mu + R\\nu),\\]\n\nand thence\n\ncoefficient \\(A'\\) \\(- \\partial A = (PA, PB, PG)(\\lambda, \\mu, \\nu) - (PA, QA, RA)(\\lambda, \\mu, \\nu),\\)\n\nwhich is\n\n\\[\\mu \\{(C\\partial_y - B\\partial_z)B - (A\\partial_z - C\\partial_x)A\\} + \\nu \\{(C\\partial_y - B\\partial_z)G - (B\\partial_x - A\\partial_y)A\\},\\]\n\nwhere coefficient of \\(\\mu\\) is\n\n\\[- A\\partial_z A - B\\partial_z B + C(\\partial_x A + \\partial_y B)\\]\n\n\\[= -(A\\partial_z A + B\\partial_z B + C\\partial_z G) = -\\frac{1}{m-1}x\\partial_z H,\\]\n\nand coefficient of \\(\\nu\\) is\n\n\\[+ (A\\partial_y A + B\\partial_y B + C\\partial_y G) = \\frac{1}{m-1}x\\partial_y H,\\]\n\nso that\n\ncoefficient \\(A'\\) \\(- \\partial A = -\\frac{1}{m-1}x(\\mu\\partial_z H - \\nu\\partial_y H).\\)\n\n53. And by forming in a similar manner the coefficients of the other terms, it appears that\n\n\\[\\text{Jac.}(U, H, \\nabla)H - (\\partial A, \\ldots)(A', B', C')^2\\]\n\n\\[-\\frac{1}{m-1}(A'x + B'y + C'z) \\begin{vmatrix}\nA' & B' & C' \\\\\n\\lambda & \\mu & \\nu \\\\\n\\partial_x H & \\partial_y H & \\partial_z H\n\\end{vmatrix},\\]\n\nor since the determinant is\n\n\\[\\begin{vmatrix}\nA' & B' & C' \\\\\n\\lambda & \\mu & \\nu \\\\\nA' & B' & C'\n\\end{vmatrix} = 0,\\]\n\nwe have the required equation,\n\n\\[\\text{Jac.}(U, H, \\nabla)H = (\\partial A, \\ldots)(A', B', C')^2.\\]\n\nThis completes the series of formulæ used in the transformations of the condition for the sextactic point.\nAPPENDIX, Nos. 54 to 74.\n\nFor the sake of exhibiting in their proper connexion some of the formulæ employed in the foregoing first transformation of the condition for a sextactic point, I have investigated them in the present Appendix, which however is numbered continuously with the memoir.\n\n54. The investigations of my former memoir and the present memoir have reference to the operations\n\n\\[ \\partial_1 = dx \\partial_x + dy \\partial_y + dz \\partial_z, \\]\n\\[ \\partial_2 = d^2x \\partial_x + d^2y \\partial_y + d^2z \\partial_z, \\]\n\\[ \\partial_3 = d^3x \\partial_x + d^3y \\partial_y + d^3z \\partial_z, \\]\n&c.,\n\nwhere if (A, B, C) are the first differential coefficients of a function \\( U = (*\\xi x, y, z)^m \\), and \\( \\lambda, \\mu, \\nu \\) are arbitrary constants, then we have\n\n\\[ dx = B\\nu - C\\mu, \\quad dy = C\\lambda - A\\nu, \\quad dz = A\\mu - B\\lambda; \\]\n\nso that putting\n\n\\[ \\partial = (B\\nu - C\\mu) \\partial_x + (C\\lambda - A\\nu) \\partial_y + (A\\mu - B\\lambda) \\partial_z \\]\n\n\\[ = \\begin{vmatrix} A & B & C \\\\ \\lambda & \\mu & \\nu \\\\ \\partial_x & \\partial_y & \\partial_z \\end{vmatrix}, \\]\n\nwe have \\( \\partial_1 = \\partial \\). The foregoing expressions of \\((dx, dy, dz)\\) determine of course the values of \\((d^2x, d^2y, d^2z), (d^3x, d^3y, d^3z), \\ldots \\), and it is throughout assumed that these values are substituted in the symbols \\( \\partial_2, \\partial_3, \\ldots \\), so that \\( \\partial_1 = \\partial \\), and \\( \\partial_2, \\partial_3, \\ldots \\) denote each of them an operator such as \\( X \\partial_x + Y \\partial_y + Z \\partial_z \\), where \\((X, Y, Z)\\) are functions of the coordinates; such operator, in so far as it is a function of the coordinates, may therefore be made an operand, and be operated upon by itself or any other like operator.\n\n55. Taking \\((a, b, c, f, g, h)\\) for the second differential coefficients of \\(U\\), \\((A, B, C, F, G, H)\\) for the inverse coefficients, and \\(H\\) for the Hessian, I write also\n\n\\[ \\Phi = (\\lambda, \\mu, \\nu)^2, \\]\n\\[ \\nabla = (\\lambda, \\mu, \\nu)(\\partial_x, \\partial_y, \\partial_z), \\]\n\\[ \\square = (\\lambda, \\mu, \\nu)(\\partial_x, \\partial_y, \\partial_z)^2, \\]\n\\[ \\delta = \\lambda x + \\mu y + \\nu z, \\]\n\\[ \\Omega = (\\lambda, \\mu, \\nu)(\\partial_x, \\partial_y, \\partial_z)^2 H = \\square H, \\]\n\\[ \\Psi = (\\lambda, \\mu, \\nu)(\\partial_x H, \\partial_y H, \\partial_z H)^2, \\]\n\\[ \\Gamma = (a, b, c, f, g, h)(\\mu \\partial_x - \\nu \\partial_y, \\nu \\partial_x - \\lambda \\partial_z, \\lambda \\partial_y - \\mu \\partial_z)^2, \\]\nand I notice that we have\n\n\\[ \\Gamma U = 2\\Phi, \\quad \\nabla U = \\frac{S}{m-1} H, \\quad \\Box U = 3H, \\]\n\n\\[ \\nabla S = \\Phi, \\quad \\nabla^2 U = H\\Phi, \\quad \\nabla \\cdot \\delta = 0, \\]\n\nthe last of which is proved, post No. 65; the others are found without any difficulty.\n\n56. I form the Table\n\n\\[ \\partial_1 U = 0, \\]\n\n\\[ \\partial_1^2 U = \\frac{mU}{m-1} \\Phi + \\frac{S^2}{(m-1)^2} (-H), \\]\n\n\\[ \\partial_2 U = \\frac{mU}{m-1} (-\\Phi) + \\frac{S^2}{(m-1)^2} (H), \\]\n\n\\[ \\partial_1^3 U = \\frac{mU}{m-1} \\partial \\Phi + \\frac{S^2}{(m-1)^2} (-\\partial H), \\]\n\n\\[ \\partial_1 \\partial_2 U = 0, \\]\n\n\\[ \\partial_3 U = \\frac{mU}{m-1} (-\\partial \\Phi) + \\frac{S^2}{(m-1)^2} (\\partial H), \\]\n\n\\[ \\partial_1^4 U = \\frac{mU}{m-1} \\left( \\partial^2 \\Phi - \\frac{2mS}{m-1} \\nabla \\Phi \\right) + \\frac{S^2}{(m-1)^2} \\left( -\\partial^2 H - \\frac{3m-6}{m-1} H\\Phi + \\frac{2S}{m-1} \\nabla H \\right), \\]\n\n\\[ \\partial_1^2 \\partial_2 U = \\frac{mU}{m-1} \\left( \\frac{1}{3} \\partial_2 \\Phi + \\frac{2S}{m-1} \\nabla \\Phi \\right) + \\frac{S^2}{(m-1)^2} \\left( -\\frac{1}{3} \\partial_2 H + \\frac{m-2}{m-1} H\\Phi - \\frac{2S}{m-1} \\nabla H \\right), \\]\n\n\\[ \\partial_1 \\partial_3 U = \\frac{mU}{m-1} \\left( -\\frac{1}{3} \\partial_3 \\Phi - \\Phi^2 - \\frac{2S}{m-1} \\nabla \\Phi \\right) + \\frac{S^2}{(m-1)^2} \\left( \\frac{1}{3} \\partial_3 H + \\frac{1}{m-1} H\\Phi + \\frac{2S}{m-1} \\nabla H \\right), \\]\n\n\\[ \\partial_2^2 U = \\frac{mU}{m-1} (\\Phi^2) + \\frac{S^2}{(m-1)^2} (-H\\Phi), \\]\n\n\\[ \\partial_4 U = \\frac{mU}{m-1} \\left( -\\frac{2}{3} \\partial_2 \\Phi - \\partial^2 \\Phi + \\Phi^2 + \\frac{2S}{m-1} \\nabla \\Phi \\right) + \\frac{S^2}{(m-1)^2} \\left( \\frac{2}{3} \\partial_2 H + \\partial^2 H - \\frac{1}{m-1} H\\Phi - \\frac{2S}{m-1} \\nabla H \\right), \\]\n\n\\[ \\partial_2 H = -\\frac{3m-6}{m-1} H\\Phi + \\frac{S}{m-1} \\nabla H, \\]\n\nand assuming \\( U = 0, \\)\n\n\\[ \\partial_1^2 H = \\partial^2 H = -\\frac{(3m-6)(3m-7)}{(m-1)^2} H\\Phi + \\frac{6m-14}{(m-1)^2} \\nabla H - \\frac{S^2}{(m-1)^2} \\Omega, \\]\n\n\\[ (\\partial_1 H)^3 = (\\partial H)^2 = -\\frac{(3m-6)^2}{(m-1)^2} H^2 \\Phi + \\frac{6m-12}{(m-1)^2} \\nabla H \\nabla H - \\frac{S^2}{(m-1)^2} \\Psi, \\]\n\nwhich are for the most part given in my former memoir; the expressions for \\( \\partial_3 U, \\partial_3 U, \\) which are not explicitly given, follow at once from the equations\n\n\\[ (\\partial_1^2 + \\partial_2^2) U = 0, \\quad (\\partial_1^3 + 2\\partial_1 \\partial_2 + \\partial_3) U = 0; \\]\n\nthose for \\( \\partial_1 \\partial_3 U, \\partial_2^2 U, \\) and \\( \\partial_4 U \\) are new, but when the expressions for \\( \\partial_1 \\partial_3 U \\) and \\( \\partial_2^2 U \\) are\nknown, that for $\\partial_4 U$ is at once found from the equation\n\n$$\\left(\\partial_1^2 + \\partial_2^2 + 4\\partial_1\\partial_3 + 3\\partial_2^2 + \\partial_4\\right)U = 0.$$\n\n57. Before going further, I remark that we have identically\n\n$$\\begin{vmatrix}\n(a, \\ldots)(x, y, z)^2(a, \\ldots)(w\\gamma - v\\beta, w\\alpha - \\lambda\\gamma, \\lambda\\beta - \\mu\\alpha)^2 \\\\\nax + hy + gz, hx + by + fz, gx + fy + cz \\\\\n\\lambda, \\mu, \\nu \\\\\n\\alpha, \\beta, \\gamma\n\\end{vmatrix}^2$$\n\n$$=(A, \\ldots)(p - \\alpha S, p - \\beta S, p - \\gamma S)^2,$$\n\n(if for shortness $p = ax + by + vz$, $S = \\lambda x + \\mu y + \\nu z$)\n\n$$= p^2(A, \\ldots)(\\lambda, \\mu, \\nu)^2$$\n\n$$- 2p\\partial(A, \\ldots)(\\lambda, \\mu, \\nu)(\\alpha, \\beta, \\gamma)$$\n\n$$+ S^2(A, \\ldots)(\\alpha, \\beta, \\gamma)^2.$$\n\n58. If in this equation we take $(a, b, c, f, g, h)$ to be the second differential coefficients of $U$, and write also $(\\alpha, \\beta, \\gamma) = (\\partial_x, \\partial_y, \\partial_z)$, the equation becomes\n\n$$m(m-1)U\\Gamma - (m-1)^2\\partial^2 = \\Phi(x\\partial_x + y\\partial_y + z\\partial_z)^2$$\n\n$$- 2S(x\\partial_x + y\\partial_y + z\\partial_z)\\nabla$$\n\n$$+ S^2\\Box,$$\n\nwhich is a general equation for the transformation of $\\partial^2(\\partial_i^2)$.\n\n59. If with the two sides of this equation we operate on $U$, we obtain\n\n$$m(m-1)U\\Gamma U - (m-1)^2\\partial^2 U = m(m-1)\\Phi U$$\n\n$$- 2(m-1)S\\nabla U$$\n\n$$+ S^2\\Box U;$$\n\nand substituting the values\n\n$$\\Gamma U = 2\\Phi, \\nabla U = \\frac{\\partial}{m-1} H, \\Box U = 3H,$$\n\nwe find the before-mentioned expression of $\\partial_i^2 U$.\n\n60. Operating with the two sides of the same equation on a function $H$ of the order $m'$, we find\n\n$$m(m-1)U\\Gamma H - (m-1)^2\\partial^2 H = m'(m'-1)\\Phi H$$\n\n$$- 2(m'-1)S\\nabla H$$\n\n$$+ S^2\\Box H;$$\n\nand in particular if $H$ is the Hessian, then writing $m' = 3m - 6$, and putting $U = 0$, we find the before-mentioned expression for $\\partial^2 H$.\n\n61. But we may also from the general identical equation deduce the expression for $(\\partial H)^2$. In fact taking $H$ a function of the degree $m'$ and writing\n\n$$(\\alpha, \\beta, \\gamma) = (\\partial_x H, \\partial_y H, \\partial_z H),$$\nwe have\n\n\\[ m(m-1)U(a, \\ldots, \\lambda \\mu \\partial_z H - \\nu \\partial_y H, \\nu \\partial_x H - \\lambda \\partial_z H, \\lambda \\partial_y H - \\mu \\partial_x H)^2 = (m-1)^2 (\\partial H)^2 \\]\n\n\\[ = m^2 \\Phi H^2 - 2m' \\Phi H \\nabla H + \\Phi^2 (A, \\ldots, \\lambda \\partial_x H, \\partial_y H, \\partial_z H)^2; \\]\n\nand if \\( H \\) be the Hessian, then writing \\( m' = 3m - 6 \\) and putting also \\( U = 0 \\), we find the before-mentioned expression for \\( (\\partial H)^2 \\).\n\n62. Proof of equation\n\n\\[ \\partial_2 = -\\frac{1}{m-1} (x \\partial_x + y \\partial_y + z \\partial_z) + \\frac{s}{m-1} \\nabla. \\]\n\nWe have\n\n\\[ \\partial_2 = \\partial \\cdot \\partial = \\{(B_y - C_y) \\partial_x + (C_x - A_x) \\partial_y + (A_y - B_y) \\partial_z \\}. \\]\n\nwhich is\n\n\\[ = \\lambda (C_y - B_y) + \\mu (A_y - C_y) + \\nu (B_y - A_y), \\]\n\nwhere\n\n\\[ A' = \\partial A = a(B_y - C_y) + h(C_x - A_x) + g(A_y - B_y) \\]\n\n\\[ = \\lambda (hC - gB) + \\mu (gA - aC) + \\nu (aB - hA), \\]\n\nwith the like values for \\( B' \\) and \\( C' \\). Substituting the values\n\n\\[ (m-1)(A, B, C) = (ax + hy + gz, hx + by + fz, gx + fy + cz), \\]\n\nwe have\n\n\\[ (m-1)A' = \\lambda (Cy - Bz) + \\mu (Fy - Ez) + \\nu (Cy - Fz); \\]\n\nand similarly\n\n\\[ (m-1)B' = \\lambda (Az - Gx) + \\mu (Gz - Fx) + \\nu (Gz - Ex), \\]\n\n\\[ (m-1)C' = \\lambda (Ex - Ay) + \\mu (Ex - By) + \\nu (Fx - Ey), \\]\n\nand then\n\n\\[ (m-1)(C_y - B_z) = \\lambda [(Ax - Ay) \\partial_y - (Az - Gx) \\partial_z] \\]\n\n\\[ + \\mu [(Bx - By) \\partial_y - (Bz - Fx) \\partial_z] \\]\n\n\\[ + \\nu [(Cx - Cy) \\partial_y - (Cx - Ex) \\partial_z] \\]\n\n\\[ = \\lambda [x(A, B, G) \\partial_x, \\partial_y, \\partial_z] - A(x \\partial_x + y \\partial_y + z \\partial_z) \\]\n\n\\[ + \\mu [x(B, B, F) \\partial_x, \\partial_y, \\partial_z] - B(x \\partial_x + y \\partial_y + z \\partial_z) \\]\n\n\\[ + \\nu [x(G, F, C) \\partial_x, \\partial_y, \\partial_z] - G(x \\partial_x + y \\partial_y + z \\partial_z)] \\]\n\nthat is\n\n\\[ (m-1)(C_y - B_z) = x \\nabla - (A, B, G)(\\lambda, \\mu, \\nu)(x \\partial_x + y \\partial_y + z \\partial_z), \\]\n\nand so\n\n\\[ (m-1)(A_y - C_y) = y \\nabla - (B, B, F)(\\lambda, \\mu, \\nu)(x \\partial_x + y \\partial_y + z \\partial_z), \\]\n\n\\[ (m-1)(B_x - A_x) = z \\nabla - (G, F, C)(\\lambda, \\mu, \\nu)(x \\partial_x + y \\partial_y + z \\partial_z); \\]\n\nwhence\n\n\\[ (m-1)\\partial_2 = (x + \\mu y + \\nu z) \\nabla - (A, \\ldots, \\lambda, \\mu, \\nu)(x \\partial_x + y \\partial_y + z \\partial_z) \\]\n\n\\[ = \\Phi(x \\partial_x + y \\partial_y + z \\partial_z); \\]\n\nor finally\n\n\\[ \\partial_2 = -\\frac{1}{m-1} \\Phi(x \\partial_x + y \\partial_y + z \\partial_z) + \\frac{s}{m-1} \\nabla. \\]\n63. This leads to the expression for $\\partial_2^2U$; we have\n\n$$\\partial_2^2 = \\frac{1}{(m-1)^2} \\Phi^2(x\\partial_x + y\\partial_y + z\\partial_z)^2$$\n\n$$- \\frac{2S}{(m-1)^2} \\Phi \\nabla(x\\partial_x + y\\partial_y + z\\partial_z)$$\n\n$$+ \\frac{S^2}{(m-1)^2} \\nabla^2;$$\n\nand operating herewith on $U$, we find\n\n$$\\partial_3^2U = \\frac{m(m-1)}{(m-1)^2} \\Phi^2U$$\n\n$$- \\frac{2(m-1)S}{(m-1)^2} \\Phi \\nabla U$$\n\n$$+ \\frac{S^2}{(m-1)^2} \\nabla^2U;$$\n\nor since\n\n$$\\nabla U = \\frac{S}{m-1} H, \\nabla^2U = H\\Phi,$$\n\nthis is\n\n$$\\partial_3^2U = \\frac{mU}{(m-1)^2} \\Phi^2 + \\frac{S^2}{(m-1)^2} H\\Phi.$$\n\n64. We have $\\partial_1\\partial_2U = 0$, and thence\n\n$$(\\partial_1\\partial_2 + \\partial_2\\partial_3 + \\partial_3\\partial_1)U = 0,$$\n\nthat is\n\n$$\\partial_3\\partial_1U = -\\partial_2\\partial_2U - \\partial_3^2U;$$\n\nor substituting the values of $\\partial_2^2U$ and $\\partial_3^2U$, we find the value of $\\partial_1\\partial_3U$ as given in the Table. And then from the equation\n\n$$(\\partial_1^4 + 6\\partial_1^2\\partial_2^2 + 4\\partial_2^4 + 3\\partial_3^2 + \\partial_4)U,$$\n\nor\n\n$$\\partial_4U = -(\\partial_1^4 + 6\\partial_1^2\\partial_2^2 + 4\\partial_2^4 + 3\\partial_3^2 + \\partial_4)U,$$\n\nwe find the value of $\\partial_4U$, and the proof of the expressions in the Table is thus completed.\n\n65. Proof of equation $\\nabla.\\partial = 0$.\n\nWe have\n\n$$\\nabla.\\partial = \\nabla.((B\\nu - C\\mu)\\partial_x + (C\\lambda - A\\nu)\\partial_y + (A\\mu - B\\lambda)\\partial_z)$$\n\n$$= \\nabla.(A(\\mu\\partial_x - \\nu\\partial_y) + B(\\nu\\partial_y - \\lambda\\partial_z) + C(\\lambda\\partial_z - \\mu\\partial_x))$$\n\n$$= \\nabla A(\\mu\\partial_x - \\nu\\partial_y) + \\nabla B(\\nu\\partial_y - \\lambda\\partial_z) + \\nabla C(\\lambda\\partial_z - \\mu\\partial_x);$$\n\nand then\n\n$$\\nabla A = (\\lambda, \\ldots, \\lambda, \\mu, \\nu)(a, h, g) = H\\lambda,$$\n\n$$\\nabla B = (\\lambda, \\ldots, \\lambda, \\mu, \\nu)(h, b, f') = H\\mu,$$\n\n$$\\nabla C = (\\lambda, \\ldots, \\lambda, \\mu, \\nu)(g, f, c) = H\\nu;$$\n\nor substituting these values, we have the equation in question.\n66. Proof of expression for $\\partial_3$.\n\nWe have\n\n$$\\partial_2 = -\\frac{1}{m-1} \\Phi(x \\partial_x + y \\partial_y + z \\partial_z) + \\frac{s}{m-1} \\nabla;$$\n\nand thence operating on the two sides respectively with $\\partial_1$, $= \\partial$, we have\n\n$$\\partial_3 = -\\frac{1}{m-1} \\{\\partial \\Phi(x \\partial_x + y \\partial_y + z \\partial_z) + \\Phi \\partial (x \\partial_x + y \\partial_y + z \\partial_z)\\}$$\n\n$$+ \\frac{1}{m-1} \\{\\partial \\Phi \\nabla + \\Phi \\cdot \\nabla\\};$$\n\nor since\n\n$$\\partial (x \\partial_x + y \\partial_y + z \\partial_z) = \\partial, \\quad \\partial \\Phi = 0,$$\n\nthis is\n\n$$\\partial_3 = -\\frac{1}{m-1} \\partial \\Phi(x \\partial_x + y \\partial_y + z \\partial_z) - \\frac{1}{m-1} \\Phi \\partial + \\frac{s}{m-1} \\partial \\cdot \\nabla.$$\n\n67. Proof of expression for $\\partial_3 H$.\n\nOperating with $\\partial_3$ upon $H$, we have at once\n\n$$\\partial_3 H = -\\frac{3m-6}{m-1} H \\partial \\Phi - \\frac{1}{m-1} \\Phi \\partial H + \\frac{s}{m-1} (\\partial \\cdot \\nabla) H.$$\n\nThe remainder of the present Appendix is preliminary, or relating to the investigation of the expressions for $\\partial_1 \\partial_2 U$ and $\\partial_1^2 \\partial_3 U$, used ante, No. 31.\n\n68. Proof of equation $\\nabla^2 \\partial U = \\Phi \\partial H - H \\partial \\Phi$.\n\nWe have identically\n\n$$(A, \\ldots)(\\lambda, \\mu, \\nu)^q(A, \\ldots)(\\partial_x, \\partial_y, \\partial_z)^q - [(A, \\ldots)(\\lambda, \\mu, \\nu)(\\partial_x, \\partial_y, \\partial_z)]^q$$\n\n$$=(abc-\\&c)(a, \\ldots)(\\nu \\partial_y - \\mu \\partial_x, \\lambda \\partial_x - \\nu \\partial_z, \\mu \\partial_x - \\lambda \\partial_y)^q;$$\n\nthat is\n\n$$\\Phi \\square - \\nabla^2 = H \\Gamma;$$\n\nand then multiplying by $\\partial$, and with the result operating on $U$, we find\n\n$$\\Phi \\square \\partial U - \\nabla^2 \\partial U = H \\Gamma \\partial U.$$\n\nNow\n\n$$\\square U = (A, \\ldots)(\\partial_x, \\partial_y, \\partial_z)^q U$$\n\n$$=(A, \\ldots)(a, b, c, 2f, 2g, 2h);$$\n\nand thence\n\n$$\\square \\partial U = (A, \\ldots)(\\partial a, \\partial b, \\partial c, 2\\partial f, 2\\partial g, 2\\partial h);$$\n\nand observing that\n\n$$H = \\begin{vmatrix} a, & h, & g \\\\ h, & b, & f \\\\ g, & f, & c \\end{vmatrix},$$\nand thence that\n\n\\[ \\partial H = \\begin{vmatrix} \\partial a, \\partial h, \\partial g \\\\ h, b, f \\\\ g, f, c \\end{vmatrix} + \\begin{vmatrix} a, h, g \\\\ \\partial h, \\partial b, \\partial f \\\\ g, f, c \\end{vmatrix} + \\begin{vmatrix} a, h, g \\\\ h, b, f \\\\ \\partial g, \\partial f, \\partial c \\end{vmatrix} \\]\n\n\\[ = (\\mathcal{A}, \\mathcal{B}, \\mathcal{C})(\\partial a, \\partial h, \\partial g) + (\\mathcal{H}, \\mathcal{F}, \\mathcal{G})(\\partial h, \\partial b, \\partial f) + (\\mathcal{E}, \\mathcal{F}, \\mathcal{C})(\\partial g, \\partial f, \\partial c) \\]\n\n\\[ = (\\mathcal{A}, \\ldots)(\\partial a, \\partial b, \\partial c, 2\\partial f, 2\\partial g, 2\\partial h), \\]\n\nwe see that\n\n\\[ \\Box \\partial U = \\partial H. \\]\n\nMoreover\n\n\\[ \\Gamma U = (a, \\ldots)(\\nu \\partial_y - \\mu \\partial_z, \\ldots)^2 U \\]\n\n\\[ = a(b^2 + c^2 - 2f\\mu) \\]\n\n\\[ + b(c^2 + a^2 - 2g\\lambda) \\]\n\n\\[ + c(a^2 + b^2 - 2h\\mu) \\]\n\n\\[ + 2f(-f\\lambda^2 + g\\lambda\\mu + h\\lambda\\mu - a\\mu) \\]\n\n\\[ + 2g(f\\lambda\\mu - g\\mu^2 + h\\mu^2 - b\\lambda) \\]\n\n\\[ + 2h(f\\lambda + g\\mu - h\\mu^2 - c\\mu); \\]\n\nand thence\n\n\\[ \\Gamma \\partial U = (a, \\ldots)(\\nu \\partial_y - \\mu \\partial_z, \\ldots)^2 \\partial U \\]\n\n\\[ = a(\\nu \\partial b + \\mu \\partial c - 2\\mu \\nu \\partial f') \\]\n\n\\[ + &c. \\]\n\n\\[ = \\lambda(b \\partial e + c \\partial b - 2f \\partial f') \\]\n\n\\[ + &c. \\]\n\n\\[ = (\\partial \\mathcal{A}, \\partial \\mathcal{B}, \\partial \\mathcal{C}, \\partial \\mathcal{F}, \\partial \\mathcal{G}, \\partial \\mathcal{H})(\\lambda, \\mu, \\nu)^2, \\]\n\nthat is\n\n\\[ \\Gamma \\partial U = \\partial \\Phi. \\]\n\nHence the equation\n\n\\[ \\Phi \\Box \\partial U - \\nabla^2 \\partial U = H \\Gamma \\partial U \\]\n\nbecomes\n\n\\[ \\Phi \\partial H - \\nabla^2 \\partial U = H \\partial \\Phi, \\]\n\nthat is,\n\n\\[ \\nabla^2 \\partial U = \\Phi \\partial H - H \\partial \\Phi. \\]\n\n69. Proof of equation \\( \\partial_1 \\partial_2^2 U = \\frac{s^2}{(m-1)^2} (\\Phi \\partial H - H \\partial \\Phi). \\)\n\nWe have\n\n\\[ \\partial_2^2 = \\frac{1}{(m-1)^2} \\Phi^2(x \\partial_x + y \\partial_y + z \\partial_z)^2 \\]\n\n\\[ - \\frac{2s}{(m-1)^2} \\Phi(x \\partial_x + y \\partial_y + z \\partial_z) \\nabla \\]\n\n\\[ + \\frac{s^2}{(m-1)^2} \\nabla^2; \\]\nand thence multiplying by $\\partial_1 = \\partial$, and with the result operating upon $U$, we find\n\n$$\\partial_1 \\partial_2^2 U = \\frac{(m-1)(m-2)}{(m-1)^2} \\Phi^2 U - \\frac{2(m-2)}{(m-1)^2} \\Phi \\partial \\nabla U + \\frac{s^2}{(m-1)^2} \\partial \\nabla^2 U.$$  \n\nBut $\\partial U = 0$, and thence also $\\nabla (\\partial U) = 0$, that is $(\\nabla \\cdot \\partial) U + \\nabla \\partial U = 0$; moreover $\\nabla \\cdot \\partial = 0$, and therefore $(\\nabla \\cdot \\partial) U = 0$, whence also $\\nabla \\partial U = 0$. Therefore\n\n$$\\partial_1 \\partial_2^2 U = \\frac{s^2}{(m-1)^2} \\partial \\nabla^2 U;$$\n\nor substituting for $\\partial \\nabla^2 U$ its value $= \\Phi \\partial H - H \\partial \\Phi$, we have the required expression for $\\partial_1 \\partial_2^2 U$.\n\n70. Proof of equation $\\partial_1 \\partial_3 U = \\frac{s^2}{(m-1)^3} \\left( (3m-6) H \\partial \\Phi + (-m+3) \\Phi \\partial H \\right) + \\frac{s^3}{(m-1)^3} \\{ -(\\partial \\cdot \\nabla) H \\}$.\n\nWe have\n\n$$\\partial_3 = -\\frac{1}{m-1} \\partial \\Phi (x \\partial_x + y \\partial_y + z \\partial_z) - \\frac{1}{m-1} \\Phi \\partial + \\frac{s}{m-1} \\partial \\cdot \\nabla,$$\n\nand thence multiplying by $\\partial_1 = \\partial^2$, and operating on $U$,\n\n$$\\partial_1 \\partial_3 U = -\\frac{m-2}{m-1} \\partial \\Phi \\partial^2 U - \\frac{1}{m-1} \\Phi \\partial^3 U + \\frac{s}{m-1} (\\partial \\cdot \\nabla) \\partial^3 U.$$\n\nTo reduce $(\\partial \\cdot \\nabla) \\partial^2 U$, we have\n\n$$\\partial (\\nabla \\partial^2 U) = \\nabla \\partial^3 U + (\\partial \\cdot \\nabla \\partial^2) U$$\n\n$$= \\nabla \\partial^3 U + [(\\partial \\cdot \\nabla) \\partial^2 + \\nabla (\\partial \\cdot \\partial^2)] U$$\n\n$$= \\nabla \\partial^3 U + (\\partial \\cdot \\nabla) \\partial^2 U + 2 \\nabla \\partial \\partial^2 U,$$\n\nand since\n\n$$\\partial_2 = -\\frac{1}{m-1} \\Phi (x \\partial_x + y \\partial_y + z \\partial_z) + \\frac{s}{m-1} \\nabla;$$\n\nmultiplying by $\\nabla \\partial$, and with the result operating on $U$, we obtain\n\n$$\\nabla \\partial \\partial_2 U = -\\frac{m-2}{m-1} \\Phi \\nabla \\partial U + \\frac{s}{m-1} \\nabla^2 \\partial U;$$\n\nor since $\\nabla \\partial U = 0$, this is\n\n$$\\nabla \\partial \\partial_2 U = \\frac{s}{m-1} \\nabla^2 \\partial U.$$\n\nHence\n\n$$\\partial (\\nabla \\partial^2 U) = \\nabla \\partial^3 U + (\\partial \\cdot \\nabla) \\partial^2 U + \\frac{2s}{m-1} \\nabla^2 \\partial U,$$\n\nthat is\n\n$$(\\partial \\cdot \\nabla) \\partial^2 U = \\partial (\\nabla \\partial^2 U) - \\nabla \\partial^3 U - \\frac{2s}{m-1} \\nabla^2 \\partial U.$$\n\nSubstituting this value of $(\\partial \\cdot \\nabla) \\partial^2 U$, we find\n\n$$\\partial_1 \\partial_3 U = -\\frac{m-2}{m-1} \\partial \\Phi \\partial^2 U - \\frac{1}{m-1} \\Phi \\partial^3 U$$\n\n$$+ \\frac{s}{m-1} \\left( \\partial (\\nabla \\partial^2 U) - \\nabla \\partial^3 U \\right)$$\n\n$$+ \\frac{s^2}{(m-1)^2} (-2 \\nabla^2 \\partial U),$$\n\nthe three lines whereof are to be separately further reduced.\n71. For the first line we have\n\n\\[ \\partial^3 U = -\\frac{s^2}{(m-1)^2} H, \\quad \\partial^3 U = -\\frac{s^2}{(m-1)^2} \\partial H, \\]\n\nand hence\n\n\\[ \\text{first line of } \\partial^3 \\partial^3 U = \\frac{s^2}{(m-1)^3} ((m-2)H \\Phi + \\Phi \\partial H). \\]\n\n72. For the second line, we have\n\n\\[ \\nabla (\\partial^3 U) = \\nabla \\partial^3 U + 2(\\nabla \\cdot \\partial) \\partial U \\]\n\\[ = \\nabla \\partial^3 U, \\text{ since } \\nabla \\cdot \\partial = 0, \\text{ and therefore } (\\nabla \\cdot \\partial) \\partial U = 0; \\]\n\nthat is\n\n\\[ \\nabla \\partial^3 U = \\nabla (\\partial^3 U) = \\nabla \\left( \\frac{mU}{m-1} \\Phi - \\frac{s^2}{(m-1)^2} H \\right) \\]\n\\[ = \\frac{m}{m-1} (U \\nabla \\Phi + \\Phi \\nabla U) - \\frac{1}{(m-1)^2} (S^2 \\nabla H + 2S \\nabla S); \\]\n\nor writing\n\n\\[ U = 0, \\quad \\nabla U = \\frac{s}{m-1} H, \\quad \\nabla S = \\Phi, \\]\n\nthis is\n\n\\[ \\nabla \\partial^3 U = \\frac{(m-2)s}{(m-1)^2} H \\Phi - \\frac{s^2}{(m-1)^2} \\nabla H, \\]\n\nwhence also\n\n\\[ \\partial (\\nabla \\partial^3 U) = \\frac{(m-2)s}{(m-1)^2} (H \\Phi + \\Phi \\partial H) - \\frac{s^2}{(m-1)^2} \\partial (\\nabla H). \\]\n\nSimilarly\n\n\\[ \\nabla \\partial^3 U = \\nabla (\\partial^3 U) \\]\n\\[ = \\nabla \\left( \\frac{mU}{m-1} \\Phi - \\frac{s^2}{(m-1)^2} \\partial H \\right) \\]\n\\[ = \\frac{m}{m-1} (U \\nabla \\Phi + U \\nabla (\\partial \\Phi)) - \\frac{1}{(m-1)^2} (S^2 \\nabla (\\partial H) + 2S \\nabla \\partial H); \\]\n\nor putting\n\n\\[ U = 0, \\quad \\nabla U = \\frac{s}{m-1} H, \\quad \\nabla S = \\Phi, \\]\n\nand observing also that \\( \\nabla (\\partial H), = \\nabla \\partial H + (\\nabla \\cdot \\partial) H \\) is equal to \\( \\nabla \\partial H \\), that is to \\( \\partial \\nabla H \\),\n\nwe obtain\n\n\\[ \\nabla \\partial^3 U = \\frac{s}{(m-1)^2} (mH \\Phi - 2 \\Phi \\partial H) - \\frac{s^2}{(m-1)^2} \\partial \\nabla H; \\]\n\nand then from the above value of \\( \\partial (\\nabla \\partial^3 U) \\), we find\n\n\\[ \\partial (\\nabla \\partial^3 U) - \\nabla \\partial^3 U = \\frac{s}{(m-1)^2} (-2H \\Phi + m \\Phi \\partial H) + \\frac{s^2}{(m-1)^2} (-\\partial (\\nabla H) + \\partial \\nabla H); \\]\n\nor observing that the term multiplied by \\( \\frac{s^2}{(m-1)^2} \\) is \\( -(\\partial \\cdot \\nabla) H \\), we find\n\n\\[ \\text{second line of } \\partial^3 \\partial^3 U = \\frac{s^2}{(m-1)^3} (-2H \\Phi + m \\Phi \\partial H) + \\frac{s^2}{(m-1)^3} (-(\\partial \\cdot \\nabla) H). \\]\n73. For the third line, substituting for $\\nabla^2 \\partial U$ its value $= \\Phi \\partial H - H \\partial \\Phi$, we have\n\nthird line of $\\partial_1^2 \\partial_3 U = - \\frac{2s^2}{(m-1)^2} (\\Phi \\partial H - H \\partial \\Phi)$.\n\n74. Hence, uniting the three lines, we have\n\n$$\\partial_1^2 \\partial_3 U = \\frac{s^2}{(m-1)^8} ((m-2)H \\partial \\Phi + \\Phi \\partial H)$$\n\n$$+ \\frac{s^2}{(m-1)^8} (-2H \\partial \\Phi + m \\Phi \\partial H) + \\frac{s^3}{(m-1)^8} (-(\\partial \\cdot \\nabla) H)$$\n\n$$+ \\frac{s^2}{(m-1)^8} ((2m-2)H \\partial \\Phi + (-2m+2)\\Phi \\partial H),$$\n\nand reducing, we have the above-mentioned value of $\\partial_1^2 \\partial_3 U$.",
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    "identifier": "jstor-108894",
    "title": "On the Sextactic Points of a Plane Curve",
    "authors": "A. Cayley",
    "year": 1865,
    "volume": "155",
    "journal": "Philosophical Transactions of the Royal Society of London",
    "page_count": 35,
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