The Taylorian Series or Law of the Mean may now be extended, under proper conditions, to develop f(x, y) near (x, y), thus : af fix + h, Y + k)=-i- (x.Y) + i h + k ax 1 1._ ay aif i χ 2hk + k j ...
1 P = + Oh, y + ny) + rn pi: ars-PyP ' Symbolically, f(x+h, h + k f(x, y) The last term is the remainder Rn, which must converge upon 0, for n 0), to be neglected. A sufficient condition therefore is that the partial D's of f remain finite near (x, y) for n. This is not a necessary condition, how ever, to find which is not easy nor attempted here.
Geometrically, take the tangent plane at P as XY, the normal as Z. Develop z== f(x, y) near P(0, 0, 0) so that ris°= x, ds==s. We have x ax + (0, 0+ 1 { a°, 0) + i az ax ay ay l 2 f Since x, y, z are all infinitesimal and a ax ay at (0, 0, 0), z is infinitesimal of 2d order. Call the 2d D's in order A, B, C and put x = r cos 8, r sin 8, where a is the angle with ZX of a normal plane, making a normal section, through Z. Hence-- =A cos 8 +2 B cos 6 sin 6 +Csin 8 s 2 Am r ,hence Lim. +A cos 6 +2B cos 6 sin +, + C sin which is easily seen to be the curvature, a= ofthis normal section. For a perpendicular normal section, 8'=8 +x/2, 2 K'== 1 sin 2B sin a cos 6 + C cos 8 ; P' 1 -- hence P + B, a constant for all pairs of perpendicular normal sections (Euler), im portant in Physics and formerly taken as measure of the curvature at P(0, 0, 0).
Consider the surface 2z = Ax'-1- 2Bxy Cy'. It is a Paraboloid (Pd); it fits on S only at P (0, 0,0), elsewhere departs from S. The sec tions of S and Pd are not the same for s=c, but close down on each other for c The Pd-section is an ellipse, an hyperbola, or a parabola, according as
AC
Hence Pd and S agree elementally at P(0, 0, 0) ; also they agree in curvature (of their own normal sections), hence Pd is called the osculating paraboloid of S at (0, 0, 0). All these parallel sections, for changing c, are similar, hence Ax'+ 2Bxy --- i 1 is taken as type and called Indicatrix (Dupin). This
indicatrix is an ellipse for B' AC <0, the S is cup-shaped or synclastic; it is an hyperbola for B'AC>O, the S is saddle-shaped (anti elastic), like a mountain-pass. The indicatrix has two Axes, tangents to sections of greatest curvature both of Pd and S at P (any point of S) which are mutually perpendicular and named principal Now let P start to move on S facing along either axis or prin cipal section (say that of least curvature). This axis starts to turn about P. Let P continue to move on S facing always along the turning axis. The tangent to its path will give the direction of this axis at every point of its path, which path is called a Line of Curvature (LC). Plainly through every point of S there pass in general two and only two LC's (Mongc), each, the envelope on S of a system of principal tangents to S. These LC's cut up S into elementary curvilinear rectangles and yield an excellent system of co-ordinates (u, v). If the indicatrix be a circle, then all its axes are principal, through the point P there pass an 'CI of LC's, every normal section is principal, the point P is an umbilic or cyclic point. If the indicatrix be a parabola, then S is edged or ridged (cylindric) at P.
The notion of surface-curvature is generated and defined quite like that of line-curvature. Draw the normal N to S at every point of the border B of dS', forming a ruled surface, R. Draw parallel to each N a radius of a unit sphere, forming a cone C cutting out 4S', on the sphere-surface, which subtends a (so-called) solid angle da at the centre. .This we also define as the solid angle of the N's, and further define the average curvature of dS as the ratio (Think of a cord passed round the gorge dS of R and then tightened, compressing R into C without changing the solid angle). If the unit solid angle or stereradian (Halsted) be subtended by r', the whole solid angle about centre=4r; then the metric numbers of da and dS' are equal, hence the curvature of 45= dS dS' dS and Lim. a. instantaneouscurvature at P (in dS) K.