% This paper has been transcribed in Plain TeX by % David R. Wilkins % School of Mathematics, Trinity College, Dublin 2, Ireland % (dwilkins@maths.tcd.ie) % % Trinity College, 2000. \magnification=\magstep1 \vsize=227 true mm \hsize=170 true mm \voffset=-0.4 true mm \hoffset=-5.4 true mm \def\folio{\ifnum\pageno>0 \number\pageno \else \ifnum\pageno<0 \romannumeral-\pageno \else\fi\fi} \font\Largebf=cmbx10 scaled \magstep2 \font\largerm=cmr12 \font\largeit=cmti12 \font\largesc=cmcsc10 scaled \magstep1 \font\tensc=cmcsc10 \font\sevensc=cmcsc10 scaled 700 \newfam\scfam \def\sc{\fam\scfam\tensc} \textfont\scfam=\tensc \scriptfont\scfam=\sevensc \pageno=0 \null\vskip72pt \centerline{\Largebf ON A NEW SYSTEM OF TWO GENERAL} \vskip12pt \centerline{\Largebf EQUATIONS OF CURVATURE} \vskip24pt \centerline{\Largebf By} \vskip24pt \centerline{\Largebf William Rowan Hamilton} \vskip24pt \centerline{\largerm (Proceedings of the Royal Irish Academy, 9 (1867), pp.\ 302--305.)} \vskip36pt \vfill \centerline{\largerm Edited by David R. Wilkins} \vskip 12pt \centerline{\largerm 2000} \vskip36pt\eject \pageno=-1 \null\vskip36pt \centerline{\Largebf NOTE ON THE TEXT} \bigskip This edition is based on the original text published posthomously in volume~9 of the {\it Proceedings of the Royal Irish Academy}. The following obvious typographical errors have been corrected:--- \smallbreak \item{} before equation~(a), a full stop (period) has been changed to a colon; \smallbreak \item{} in equation~(k), `$(Z - x)$' has been corrected to `$(Z - z)$'; \smallbreak \item{} in equation~(o), `$C = e E'' - e' E$' has been corrected to `$C = e E' - e' E$'; \smallbreak \item{} equation~(q) in the original text was given as $$(e R^{-1} - e K^{-1}) (e'' R^{-1} - e'' K^{-1}) = (e' R^{-1} - e K^{-1})^2;$$ \smallbreak \item{} in equation~(r), `$ = 0$' has been appended to the polynomial `$R^{-2} - F R^{-1} + G$'; \smallbreak \item{} in equation~(w), the equality sign $=$ has been added. \bigbreak\bigskip \line{\hfil David R. Wilkins} \vskip3pt \line{\hfil Dublin, March 2000} \vfill\eject \pageno=1 \null\vskip36pt \centerline{\sc On a New System of Two General Equations of Curvature,} \bigskip \begingroup \advance\leftskip by 24pt \rightskip=\leftskip \parindent=-\leftskip Including as easy consequences a new form of the Joint Differential Equation of the Two Lines of Curvature, with a new Proof of their General Rectangularity; and also a new Quadratic for the Joint Determination of the Two Radii of Curvature: all deduced by Gauss's Second Method, for discussing generally the Properties of a Surface; and the latter being verified by a Comparison of Expressions, for what is called by him the Measure of Curvature. \par\endgroup \nobreak\vskip12pt \centerline{\largerm Sir William Rowan Hamilton} \nobreak\vskip12pt \centerline{Communicated June 26, 1865.} \nobreak \vskip12pt \centerline{[{\it Proceedings of the Royal Irish Academy}, vol.~ix (1867), pp.~302--305.]} \bigbreak 1. Notwithstanding the great beauty and importance of the investigations of the illustrious {\sc Gauss}, contained in his {\it Disquisitiones Generales circ\`{a} Superficies Curvas}, a Memoir which was communicated to the {\it Royal Society of G\"{o}ttingen\/} in October, 1827, and was printed in Tom.~vi. of the {\it Commentationes Recentiores}, but of which a Latin reprint has been since very judiciously given, near the beginning of the Second Part (Deuxi\`{e}me Partie, Paris, 1850) of {\sc Liouville's} {\it Edition\/}\footnote*{The foregoing dates, or references, are taken from a note to page~505 of that Edition.} {\it of\/} {\sc Monge}, it still appears that there is room for some not useless Additions to the Theory of {\it Lines\/} and {\it Radii of Curvature}, for {\it any given Curved Surface}, when treated by what Gauss calls the {\it Second Method\/} of discussing the {\it General Properties of Surfaces}. In fact, the {\it Method\/} here alluded to, and which consists chiefly in treating the {\it three\/} co-ordinates of the {\it surface\/} as being so many {\it functions\/} of {\it two\/} independent variables, does not seem to have been used {\it at all\/} by Gauss, for the determination of the {\it Directions of the Lines of Curvature\/}; and as regards the {\it Radii of Curvature\/} of the {\it Normal Sections\/} which {\it touch\/} these {\it Lines\/} of Curvature, he appears to have employed the {\it Method, only for the Product}, and {\it not also\/} for the {\it Sum}, of the {\it Reciprocals}, of those {\it Two Radii}. \bigbreak 2. As regards the {\it notations}, let $x$,~$y$,~$z$ be the rectangular co-ordinates of a point~${\sc p}$ upon a surface $(S)$, considered as {\it three\/} functions of {\it two\/} independent variables, $t$ and $u$; and let the 15 partial derivatives, or 15 partial differential coefficients, of $x$,~$y$,~$z$ taken with respect to $t$ and $u$, be given by the nine differential expressions: $$\left\{ \eqalign{ dx &= x' \, dt + x_\prime \, du;\cr dy &= y' \, dt + y_\prime \, du;\cr dz &= z' \, dt + z_\prime \, du;\cr} \quad \eqalign{ dx' &= x'' \, dt + x_\prime' \, du;\cr dy' &= y'' \, dt + y_\prime' \, du;\cr dz' &= z'' \, dt + z_\prime' \, du;\cr} \quad \eqalign{ dx_\prime &= x_\prime' \, dt + x_{\prime\prime} \, du;\cr dy_\prime &= y_\prime' \, dt + y_{\prime\prime} \, du;\cr dz_\prime &= z_\prime' \, dt + z_{\prime\prime} \, du.\cr} \right. \leqno {\rm (a)} \, . \, .$$ \bigbreak 3. Writing also, for abridgment, $$e = x'^2 + y'^2 + z'^2;\quad e' = x' x_\prime + y' y_\prime + z' z_\prime;\quad e'' = x_\prime^2 + y_\prime^2 + z_\prime^2 \leqno {\rm (b)} \, . \, .$$ we shall have $$e e'' - e'^2 = K^2, \leqno {\rm (c)} \, . \, .$$ if $$K^2 = L^2 + M^2 + N^2, \leqno {\rm (d)} \, . \, .$$ and $$L = y' z_\prime - z' y_\prime;\quad M = z' x_\prime - x' z_\prime;\quad N = x' y_\prime - y' x_\prime; \leqno {\rm (e)} \, . \, .$$ so that $$L x' + M y' + N z' = 0,\quad L x_\prime + M y_\prime + N z_\prime = 0. \leqno {\rm (f)} \, . \, .$$ Hence $K^{-1} L$, $K^{-1} M$, $K^{-1} N$ are the {\it direction-cosines\/} of the {\it normal\/} to the surface $(S)$ at ${\sc p}$; and if $x$,~$y$,~$z$ be the co-ordinates of any {\it other\/} point~${\sc q}$ of the same normal,we shall have the equations $$K (X - x) = LR;\quad K (Y - y) = MR;\quad K (Z - z) = NR; \leqno {\rm (g)} \, . \, .$$ with $$R^2 = (X - x)^2 + (Y - y)^2 + (Z - z)^2; \leqno {\rm (h)} \, . \, .$$ where $R$ denotes the normal line ${\sc p} {\sc q}$, considered as changing sign in passing through zero. \bigbreak 4. The following, however, is for some purposes a more convenient {\it form\/} (comp.~(f)) of the {\it Equations of the Normal\/}; $$(X - x) x' + (Y - y) y' + (Z - z) z' = 0; \leqno {\rm (i)} \, . \, .$$ $$(X - x) x_\prime + (Y - y) y_\prime + (Z - z) z_\prime = 0. \leqno {\rm (j)} \, . \, .$$ Differentiating these, as if $X$, $Y$, $Z$ were constant, that is, treating the point~${\sc q}$ as an intersection of two consecutive normals, we obtain these two other equations, $$\left\{ \eqalign{ (X - x) \, dx' + (Y - y) \, dy' + (Z - z) \, dz' &= x' \, dx + y' \, dy + z' \, dz;\cr (X - x) \, dx_\prime + (Y - y) \, dy_\prime + (Z - z) \, dz_\prime &= x_\prime \, dx + y_\prime \, dy + z_\prime dz.\cr} \right. \leqno {\rm (k)} \, . \, .$$ If, then, we write, for abridgment, $$\left\{ \eqalign{ v &= du : dt;\cr E' &= L x_\prime' + M y_\prime' + N z_\prime';\cr} \quad \eqalign{ E &= L x'' + M y'' + N z'';\cr E'' &= L x_{\prime\prime} + M y_{\prime\prime} + N z_{\prime\prime};\cr} \right. \leqno {\rm (l)} \, . \, .$$ we shall have, by (a) (b) (g), the two important formul{\ae}: $$R (E + E' v) = K (e + e' v);\quad R (E' + E'' v) = K (e' + e'' v); \leqno {\rm (m)} \, . \, .$$ which we propose to call the two general {\it Equations of Curvature}. \bigbreak 5. In fact, by elimination of $R$, these equations (m) conduct to a {\it quadratic in\/} $v$, of which the roots may be denoted by $v_1$ and $v_2$, which first presents itself under the form, $$(e + e' v) (E' + E'' v) = (e' + e'' v) (E + E' v), \leqno {\rm (n)} \, . \, .$$ but may easily be thus transformed, $$\left\{ \eqalign{ & Av^2 - Bv + C = 0, \hbox{ or } A \, du^2 - B \, dt \, du + C \, dt^2 = 0,\cr & \hbox{with } A = e' E'' - e'' E',\quad B = e'' E - e E'',\quad C = e E' - e' E;\cr} \right. \leqno {\rm (o)} \, . \, .$$ so that we have the following {\it general relation}, $$e A + e' B + e'' C = 0, \leqno {\rm (p)} \, . \, .$$ (of which we shall shortly see the geometrical signification), between the {\it coefficients}, $A$, $B$, $C$, of the {\it joint differential equation\/} of the system of the two {\it Lines of Curvature\/} on the surface. \bigbreak 6. The root $v_1$ of the quadratic (o) determines the {\it direction\/} of what may be called the {\it First Line of Curvature}, through the point~${\sc p}$ of that surface; and the {\it First Radius of Curvature}, for the same point~${\sc p}$, or the radius $R_1$ of curvature of the {\it normal section\/} of the surface which {\it touches\/} that {\it first line}, may be obtained from {\it either\/} of the two equations (m), as the value of $R$ which corresponds in that equation to the value~$v_1$ of $v$. And in like manner, the {\it Second Radius of Curvature\/} of the same surface at the same point has the value $R_2$, which answers to the value~$v_2$ of $v$, in each of the same two {\it Equations of Curvature\/} (m). We see, then, that this {\it name\/} for those two equations is justified by observing that when the two independent variables $t$ and $u$ are given or known; and therefore also the seven functions of them, above denoted by $e$,~$e'$,~$e''$, $E$,~$E'$,~$E''$, and $K$. The equations (m) are satisfied by {\it two\/} (but {\it only two\/}) {\it systems of values}, $v_1$,~$R_1$, and $v_2$,~$R_2$, of (I.) the {\it differential quotient\/}~$v$, or $\displaystyle {du \over dt}$, which determines the {\it direction\/} of a {\it line of curvature\/} on the surface; and (II.) the symbol~$R$, which determines (comp.\ No.~4) at once the {\it length\/} and the {\it direction}, of the {\it radius of curvature}, corresponding to that {\it line}. \bigbreak 7. Instead of eliminating $R$ between the two equations (m), we may {\it begin\/} by eliminating~$v$; a process which gives the following quadratic in $R^{-1}$ (the curvature):--- $$(e R^{-1} - E K^{-1}) (e'' R^{-1} - E'' K^{-1}) = (e' R^{-1} - E' K^{-1})^2; \leqno \phantom{\hbox{or}\quad} {\rm (q)} \, . \, .$$ $$R^{-2} - F R^{-1} + G = 0; \hbox{ where (because $e e'' - e'^2 = K^2$),} \leqno \hbox{or}\quad {\rm (r)} \, . \, .$$ $$F = R_1^{-1} + R_2^{-1} = (e E'' - 2 e' E' + e'' E) K^{-3}, \hbox{ and} \leqno \phantom{\hbox{or}\quad} {\rm (s)} \, . \, .$$ $$G = R_1^{-1} R_2^{-1} = (E E'' - E'^2) K^{-4}. \leqno \phantom{\hbox{or}\quad} {\rm (t)} \, . \, .$$ We ought, therefore, as a {\it First General Verification}, to find that this last expression, which may be thus written, $$G = R_1^{-1} R_2^{-1} = {E E'' - E' E' \over (L^2 + M^2 + N^2)^2}, \leqno {\rm (u)} \, . \, .$$ agrees with that reprinted in page~521 of Liouville's Monge, for what Gauss calls the {\it Measure of Curvature\/} ($k$) of a {\it Surface\/}; namely, $$k = {D D'' - D' D' \over (AA + BB + CC)^2}; \leqno {\rm (v)} \, . \, .$$ which accordingly it evidently does, because our symbols $L$~$M$~$N$ $A$~$B$~$C$ represent the combinations which he denotes by $A$~$B$~$C$~$D$~$D'$~$D''$. \bigbreak 8. As a {\it Second General Verification}, we may observe that if $I$ be the {\it inclination\/} of any {\it linear element}, $du = v \, dt$, to the {\it element\/} $du = 0$, at the point~${\sc p}$, then $$\tan I = {Kv \over e + e' v}; \leqno {\rm (w)} \, . \, .$$ and therefore, that if $H$ be the {\it angle\/} at which the {\it second crosses the first}, of {\it any two lines\/} represented {\it jointly\/} by such an equation as $$A v^2 - B v + C = 0, \hbox{ with $v_1$ and $v_2$ for roots, then} \leqno {\rm (x)} \, . \, .$$ $$\tan H = \tan (I_2 - I_1) = {K (B^2 - 4AC)^{1 \over 2} \over e A + e' B + e'' C}; \leqno {\rm (y)} \, . \, .$$ so that the {\it Condition of Rectangularity\/} ($\cos H = 0$), for any {\it two\/} such lines, may be thus written: $$e A + e' B + e'' C = 0. \leqno {\rm (z)} \, . \, .$$ But this {\it condition\/} (z) had already occurred in No.~5, as an equation (p) which is satisfied generally by the {\it Lines of Curvature\/}; we see therefore anew, by this analysis, that those {\it lines\/} on {\it any surface\/} are in general (as is indeed well known) {\it orthogonal\/} to each other. \bigbreak 9. Finally, as a {\it Third General Verification}, we may assume $x$ and $y$ {\it themselves\/} (instead of $t$ and $u$), as the two independent variables of the problem, and then, if we use {\it Monge's Notation\/} of $p$, $q$, $r$, $s$, $t$, we shall easily recover all his leading results respecting {\it Curvatures of Surfaces}, but by transformations on which we cannot here delay. \bye