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u'= 0, vʻ= 0, w'= 0, P=; p having the same meaning as before. Then the equation for determining the radii of maximum and minimum curvature becomes
pa - (a + a' + 48)
(3) Let the equation to the surface be
Xyx = ms.
U = 0, v = 0, w = 0,
u' = x, v=y, w' = %. Substituting these values in the general equation for the radii of curvature, it becomes
p® + 2 (200 + y2 + sop)
Let the curve be the helix, the equations to wbich are
ds. (2x + x)? Substituting these values in the expression for the radius of curvature, we find after certain reductions
1 (10r + 3x)!
P (21 + x) The lines of curvature at any point of a surface are found by combining the equation to the surface with the equation U(dVdx-dWdy) + V(dWdx-dUdz) + WdUdy - d V dx) = 0, U, V, W having the same meanings as before.
Between this equation and the equation to the surface and its differential we can eliminate each of the variables and its differential in succession, and thus obtain the differential equations to the projections of the lines of curvature on the co-ordinate planes.
(7) Let the surface be the ellipsoid
(1) The lines of curvature are determined by combining this with the equation (b? – cé) ædydx + (c? – a®) ydxda + (a” – b?) xdxdy = 0. (2)
, and substitute
To eliminate x and dx, multiply by the values
when we obtain
a-c? 22 x y la a
- xy=0, as the differential equation of the projection of the lines of curvature on the plane of xy.
Mr Leslie Ellis* has found a symmetrical integral of the equation representing the lines of curvature in an ellipsoid, which I shall introduce in this place, though it more properly belongs to another branch of our subject.
If in equation (2) we put
u + v + w = 1. Differentiating (3), and observing that
62 - c? + C - a* + ao – b* = 0, we get (b-c)ud(dv du)+(cʻ-a*)vd(dw du)+(a? – b*)wd(du dv)=0. (5)
* Cambridge Mathematical Journal, Vol. 11. p. 133. See also on this subject a paper by Mr Thomson in the same Journal, Vol. iv. p. 279.
f, g, h being constants. But from (4) we have
du + dv + dw = 0; and from (6) du = fdudvdw, dv = gdudvdw, dw = h dudvdw. Hence
f + g + h = 0, establishing a relation between f, g, h.
Now equation (6) implies two linear equations connecting u, v, w. Therefore a particular solution of (3) is two linear equations connecting the three variables, but the given equation (4) is linear, and therefore the solution in question is the one congruent to the problem. The other linear equation is found by eliminating the differentials from (3) by means of (6). The result is
This is evidently the equation to a cone of the second degree, having its vertex in the centre of the ellipsoid ; and the lines of curvature are determined by the intersection of this cone with the ellipsoid.
(8) Let the surface be the paraboloid
The general differential equation to the lines of curvature will be found by combining this with
(a' - a) dzdy + 2ydxdx – 2zdy dx = 0. Multiplying by z and eliminating that variable and its differential, we obtain for the differential equation of the projections of the lines of curvature on the plane of xy,
The equation to the projection on yx is
(9) Let the equation to the surface be
wYX = m.
Substituting these values in the general equation to lines of curvature, we find after some reductions,
w (y* - %*) dydx + y (x2 – m?) dxdx + x (x2 – yo) dx dy = 0, which combined with the equation to the surface gives the lines of curvature.
(10) Dupin in his Développements de Géométrie, p. 322, has demonstrated the following very remarkable theorem relative to the lines of curvature on surfaces: “If there be three systems of surfaces which intersect each other at right angles, any two of them will trace on the third its lines of curvature.”
Let the three systems of surfaces be represented by the equations f (x, y, x) = C, (1) fi (x, , )=C1, (2) f(x, y, z) = C2, (3) C, C1, C, being the variable parameters by which each individual in each system is distinguished.
If we represent the differentials of these equations taken with respect to x, y, x by U, V, W, the conditions for the surfaces intersecting at right angles are
UU, + VV, + WW, = 0,