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which, a and b being supposed to be both positive, can only represent the axis of z. The cone in this case degenerates into a straight line; and as x can never be positive, since that renders x and y impossible, it appears that the point under consideration is a cusp. The surface surrounds the negative axis of %, which it touches at the origin, so that its form resembles the shape of the flower of the convolvulus.

If a and b be of contrary signs the equation to the locus of the tangent lines is

ari ' = 0, which represents two planes perpendicular to the plane of X, y.

(4) Let the surface be the cono-cuneus of Wallis, the equation to which is

a'y(c – z*) = 0.
Here U = - 2x (c? – xo), V = 2a*y* W = 2 x z.

These all vanish when x = 0, y = 0 independently of the value of x; hence the axis of is a locus of singular points or a singular line.

u = – 2 (C? – z), v = 2a, w = 2x?,
u = 0,

Ó' = 4x2, w' = 0.
The equation to the tangent lines becomes in this case

ay? – (c? – z?) x 2 = 0, where x', y' are accentuated to distinguish them from 2, the undetermined co-ordinate of the point of contact. The preceding equation is equivalent to those of two planes perpendicular to the plane of xy,

ay' + (c? – z?)x' = 0,

ay' - (co – z?)? x' = 0. By assigning different values to z we obtain different equations corresponding to successive points taken along the axis of z.

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the preceding expressions will vanish, and therefore the line determined by these equations, and the equation to the surface is a locus of singular points. This line is the intersection of the surface by the cylinder

que + y= a*, and is evidently the generating helix. Since in the equation to the surface a* + y can never be less than a’, it appears that no part of the surface lies within the helix, which is therefore truly an edge of regression.

On proceeding to the second differential coefficients, and substituting in them the critical values of æ and y we find, retaining only the terms which become infinite from involving (x2 + y2 - a®)} in the denominator, (u) = – 2 sin 27* cos ***, (v) = 2 sin 27 * cos 4*, (w) = 0,

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so that the equation to the locus of the tangent lines is

(y"? - ??) xy + x'y (x2 - y) + 27' (rox + yy) = 0, where the accentuated letters are the current co-ordinates of the tangents, and the unaccentuated the undetermined coordinates of the point of contact. This equation may be decomposed into two factors,

y' x - x'y + 27 3 = 0,

d' x + y'y = 0, which are the equations to two planes.

Umbilici*. These are points at which the two principal radii of curvature are equal. The conditions for determining them are

1 + pé pq 1 + q*

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which are therefore the co-ordinates of four umbilici.

(7) Let the surface be the paraboloid

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• The reader is referred to Gregory's Solid Geometry for a symmetrical method of deterinining Umbilici.

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Now if a > a the value of y is possible, and there are two umbilici, the co-ordinates of which are

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(8) In the surface, the equation to which is

Xyx = m", there is an umbilicus at the point

x = M, y = m, z = m.

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