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402.] Of all radii of curvature of normal sections, to determine whether any, and what, are maxima and minima.
For all normal sections passing through a given point Q, u, v, w, H, K, L being, or involving, the partial derivedfunctions of the equation to the surface at the point, are constant, and /, m, n vary. Hence we have to determine the maximum and minimum of (31), having given the conditions
u/+ vm + wn = 0, (32) p + ,»3 + ni - l; (33)
which last results are identical with (26) and (27); whence it follows, that of all normal sections the principal ones are those whose curvature is respectively a maximum and a minimum. Hence the normal sections of greatest and least curvature are at right angles to each other.
403.] The radius of curvature of any normal section may be expressed as follows, in terms of the radii of curvature of the principal normal sections.
The proposition having been discovered by Euler, is known by the name of Euler's Theorem of Normal Sections.
Let p\ and p2 be the principal radii of curvature, and li w»i «j,
/2»»2»Ji be the direction-cosines of the elements of the lines of curvature passing through the given point on the surface, and therefore of the directions of the principal normal sections at the point; let p be the radius of curvature of any other normal section, and Imn the direction-cosines of its element; and let a be the angle between its plane and that of the maximum principal radius of curvature; then we have
vl + Vim + w» = 0, T
c/i -f vml + w»! = 0, L (38)
vl% + vm2 + wnj = 0;J
Hi + mnii + ntii — cos o, )
hh + mi*rh + = 0- (40) Also if we suppose the lines of curvature, and the normal of the surface, to constitute a system of rectangular axes at the point under consideration as the origin, we have
I = li cos a + ^ sin a, ")
the last term of the general formula for such cosines vanishing,
whence multiplying severally by /, To, n, and adding, and by reason of (38),
and multiplying the first of (43) by cos a, and the second by sin a, and adding and reducing by means of (41), we have Price, Vol. i. 4 E
H/j + K»i2 + L»! = — (cos a)2 H (sin a)2; (44)
and therefore by (31),
1 (cos a)2 (sin a)2
- = + - 5 (45)
P Pi Pi
which is Eider's Theorem; and is of importance, as by it the
radius of curvature of any normal section is expressed in terms
of the principal radii of curvature at the point in question.
404 ] Hence we have the following proposition as to the radii of curvature of any two normal sections which are perpendicular to each other.
Let p and p be the radii of curvature of two normal sections perpendicular to each other; then
1 (cos a)2 (sin a)2 1 (sin a)2 (cos a)2 — = 1 , — = 1 ,
P P\ Pi P Pi Pt
... >+i =i_ + ±; m
P P Pi P2
that is, the sum of the curvatures of any two normal sections perpendicular to each other is constant.
405.3 As an example of the preceding formulae, let us take the ellipsoid whose equation is
a2(pp-a2)n b2(pp-b2) c2(pp-c2)
the roots of which quadratic equation are the greatest and least principal radii of curvature at any point on the ellipsoid.
And as the last term of the quadratic when written in an integral form is —-j—, it follows that the product of the greatest and least radii of curvature is invariable for all points for which p is constant.
Again, putting (48) in the form 30*^
and subtracting it from the equation to the ellipsoid, we have
x2 y2 z2 _ j
a2—pp b2—pp c2—pp'
which is the equation to a concentric and confocal surface of the second order.
Also by means of (26), the directions of the principal normal sections may be determined at any point of the ellipsoid.
406.] We proceed now to consider certain singular values of Pi and p2, and the nature of a point on a surface whereat the singular values exist.
In equations (11) and (12) an ambiguity of sign exists, which is introduced in extracting the root of u2-f v2 + w2, and therefore p may be affected with a + or a — sign; and the same ambiguity of sign continues in (27).
As u, v, w are the same for different normal sections at the same point, and as p is an absolute length of line, it appears from (11) that the change of sign arises from £—x, tj—y, £—z; and therefore the change of sign implies, that the centres of curvature are for different normal sections situated on different sides of the surface.
With respect to (45) it is to be borne in mind, that pi and p% are both taken with the positive sign, and that Q has the same sign, viz. +, in both. From (45) therefore it follows, that if Pi and p2 have the same sign, p has always the same sign as either of them; and that therefore all normal sections have their curvature in the neighbourhood of the point turned in the same direction. The analytical condition derived from equation (27) that this should be the case, is, that
U2kl + V2lh + waHK must be positive. (49)
Also it is manifest that, as pi and p2 are a maximum and a minimum value of the radii of curvature, p always lies between them.
Again, if the signs of pl and p2 are different, that is, if
U'kl + V2lh +w2hk is negative, (50)
the radii of curvature of some normal sections are turned in a direction contrary to that of others, and (45) becomes
1 _ (cos a)2 _ (sin a)2 ^
P Pi Pi
let a' be such that tana' = (--) , (52)
then for all values of a, from — a' to + a', and from It — a to ir + a', the radii of curvature of normal sections are turned in the same direction; and when a = + a, and = v + a, p = 00; then the normal section becomes a straight line in its consecutive elements which abut at the point, or the curvature is suspended; see Art. 383; and for all values of a outside those limits, the curvature of the normal sections is turned in a contrary direction.
According to our hypothesis pi is the maximum and p» is the minimum radius of curvature.
In the case in which one of the principal radii of curvature
is infinite, say p, = 00 , . ,
P = , (53)
the analytical condition of which derived from (27) is
U'kl + V'lh +w2hk = 0, (54)
and which, when expanded, becomes identical with that determined in Art. 360, equation (62), as the differential equation of developable surfaces. Hence we have the geometrical meaning of this equation. One of the principal normal sections of the surface lies along the straight generating line, and therefore the curvature of this section vanishes.
Again, suppose the two values of pi and pa to be equal and of opposite signs; then the coefficient of the second term of the quadratic (27) must be equal to zero, whereby we have
U*(K-f L) + V2(M H) + Wl(H + K) 0. (55)
In this case, a of (52) = 45°; and therefore of the surface