not contain the normal line at the first point; in which case we must then apply the results of Art. 375 to determine the radius of absolute curvature. This is the case of oblique section, the osculating plane of the two elements being inclined at a given angle to the normal of the surface.

398.] I propose first to investigate the curvature of principal normal sections.

Let the equation to the surface be

and for the sake of abbreviating the notation let us employ the following symbols, as in Art. 332, and in Art. 360,

Then the equations to the normal at the point x, y, z are,

(3)

ρ

where p is the distance between (x, y, z) and (, 7, 8). Hence we have

$− = U,

n-y=vQ,

(− z = wQ. (5)

Now when two consecutive normals intersect, (5) must consist with their differentials, when x, y, z, and therefore when u, v, w, o vary; accordingly, differentiating, we have

dx + v do + 2 dv = 0,
dy + v do + adv = 0,

dz + w do + odw = 0;

whence, eliminating o and do, we have

(6)

(vdw - w dv) dx + (w du – u dw) dy + (v dv — v dv)dz = 0; (7) and as this is independent of έ, n, it is true for any point on the surface, and is therefore the differential equation of lines drawn on the surface, which are such that the consecutive normals to the surface along them intersect. These lines are called

the lines of curvature, to the consideration of which we shall hereafter return. The geometrical meaning of (7) is, that the line of intersection of the two tangent planes drawn at the extremities of ds, the element of a line on the surface, is at right angles to the tangent line whose direction-cosines are proportional to dx, dy, dz.

The equation to the lines of curvature may also be expressed as follows: since

v dw - w dv = v (v'dx + u'dy + w dz) — w (w'dx + v dy +u'dz), = (vv-ww') dx + (vu' — wv)dy + (vw-wu') dz;

and since, similarly,

wdu - u dw = (wu — vv') dx + (ww' — vu') dy + (wv′ — vw) dz,

v dv - v dv (v w' — vu) dx + (vv — vw') dy + (vu' — vv′) dz,

=

the equation to the lines of curvature becomes

(vv′ — ww') dx2 + (ww' — vu') dy2 + (vu' — vv′) dz2

+ (wv' — vw + uv-vw') dy dz + (v w' — vu + vw — wu') dz dx

And as this is of two dimensions in terms of da, dy, dz, there are in general two distinct lines of curvature passing through every point on a surface; these we shall shew to be perpendicular to each other at the point.

399.] Let therefore, as in (4), p be the length of the radius of curvature of a principal normal section; and let έ, n, be the coordinates to its extremity, so that

p2 = (§ −x)2 + (n−y)2 + (5− 2)2 ;

then as έ, n, Š, p are the same for the next consecutive normal, and therefore for the next two consecutive points, we have

(−x) dx + (n − y) dy + (5—z) dz = 0,
(§—x) d3x+(n− y) d2y + (5—z) d2z = ds2;

and as the centre of curvature is on the normal,

(9)

(10)

૨ v d2x+vd2y+wd2zi (11)

Q ds2
Ud2x+vd3y+wd2z'

(12)

Now differentiating the equation to the surface twice, we have

v d2x + v d2y + w d2z+u dx2 + v dy2 + w dz2

+2 {u'dy dz+v'dz dx+w'dx dy} = 0, (13)

and employing l, m, n, to represent the direction-cosines of the element of the curve whose curvature we are investigating, so

we have, neglecting the ambiguity of sign,

૨ = ul2+vm2 + wn2 + 2 u'mn+2v'nl+2w'lm.

Ρ

(15)

Suppose that at the point on the surface neither u, v, nor w vanishes, then this expression admits of the following modification; since

v da + v dy + w dz = 0;

(16)

(17)

(18)

But as the section whose curvature we are considering has an element coincident with that of a line of curvature, we may introduce this condition, and modify (20) accordingly; returning to (7) and (8),

vdww dv (vv' — ww') dx + (vu' — wv) dy + (vw-wu') dz; whence, by means of (16) and (19),

whereby the differential equation to the lines of curvature becomes,

U (KL) dy dz + v(L — H) dz dx + w (H − K) dx dy

or

U (KL)mn + V(LH)nl + W(HK) lm = 0.

(25)

whence, eliminating u, v, w in order, and simplifying the result by means of (20), we have

a quadratic equation, which gives the two values of p corresponding to the two radii of curvature of the principal normal sections.

If however at the point of the surface under investigation either u, v or w vanishes, the process, by which H, K, L have been formed, must be modified accordingly; and neither can the equation of the principal radii of curvature be expressed in the forms (20) and (27), nor the equation of the lines of curvature in the form (25). In this case then we are obliged to recur to the original forms, which are also most general, viz. (8) and (15).

400.] Equations * (26) give us also values for l, m, n which are the direction-cosines of the lines of curvature, and therefore of the principal normal sections; whereby it may be shewn, that the principal normal sections are at right angles to each other.

For let mini, la m2 në be the direction-cosines of the normal sections corresponding to which the radii of curvature are

*For this Article, as well as for a great part of the preceding one, and for Art. 402, I am indebted to Gregory's Solid Geometry, Cambridge, 1845.

P1, P2; then we have, writing in (27) each value of p, and subtracting one from the other,

Hence, if p1 and pa are unequal, as they will usually be, the second factor must be equal to 0; and therefore by reason of (26),

(29)

and therefore the principal normal sections intersect each other at right angles.

Hence also the lines of curvature at any point of a surface cut each other at right angles.

The radii of curvature of the principal normal sections are called the principal radii of curvature.

401.] To determine the radius of curvature of any normal section.

Observing the mode of determining the radius of curvature of such a section which was mentioned in Art. 397, it appears that we have to find the point of intersection of the normal whose equations are

and the plane which is perpendicular to the second element, and whose equation is given in (10); whence, using the same notation and the same method of investigation as that of Art. 399, we have

Q

P

= ul2+vm2 + w n2 + 2 umn+2v'nl+2w'lm;

and if neither u, v, nor w vanishes, then

(30)

= Hl2 + Km2 + Ln2,

P

(31)

H, K, L being given in equations (19), and l, m, n being the direction-cosines of the element of the curve at the point on the surface, and the curvature of which we are examining.