Now if we assume y = 0, r2 = b2 which involve

these values will satisfy the equation to the surface, and will also make U, V, and W vanish: hence, as the double signs of x and may be combined in four different ways, there are four singular points on the surface. To obtain the equation to the tangent cone we must find the values of u, v, w, u, v', w', at the singular points. These are readily seen

·y2+

b2 - c2

2

+

a2 + c2

4 a2 c2 a2 - b2 { (a2 — b3) (b* — c2)} ac

= 0.

The existence of these singular points in the wave-surface was first pointed out by Sir W. Hamilton.

(3) Let the equation to the surface be

≈ (x2 + y2 + x2) + a x2 + by3

U = 2x (≈ + a), V = 2y (≈ + b),

= 0,

W = x2 + y2 + 3×3.

8 =

At the origin where x = 0, y = 0, 0, these three quantities vanish, therefore there is a singular point at the

x.

which, a and b being supposed to be both positive, can only represent the axis of . The cone in this case degenerates into a straight line; and as ≈ can never be positive, since that renders & 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

ax2 - by2 = 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

Here U = − 2 x (c2 – z), V = 2a2y W=2a2.

z.

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

The equation to the tangent lines becomes in this case

where ', ' are accentuated to distinguish them from, the undetermined co-ordinate of the point of contact. The pre

ceding equation is equivalent to those of two planes perpendicular to the plane of xy,

By assigning different values to we obtain different equations corresponding to successive points taken along the axis of z.

(5) The equation to the héliçoide dévelopable is

a

(x2 + y2 — a2) 3

a

=

a.

0, we find as in a previous

x (x cos y sin 0)

a (x2 + y2 — a2) §

;

But we found before that

x cos y sin 0 = (x2 + y2 − a2)1 ;

;

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

x2 + y2 = a2,

and is evidently the generating helix. Since in the equation to the surface a+y can never be less than a2, 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 a and y we find, retaining only the terms which become infinite from involving (x2 + y2 − a2)3 in the denominator,

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

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,

which are the equations to two planes.

Umbilici*. These are points at which the two principal

Substituting these values in the conditions for an umbilicus,

which are therefore the co-ordinates of four umbilici.

(7) Let the surface be the paraboloid

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

In order that these equations may hold we must have either = 0, or y = 0. Taking the former we find

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

a -a
a'

x = 0,

y = ± } {a' (a - a') } 3,

≈=

4.

If a <a' we must take y = 0, and then we find

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

y = m*,

there is an umbilicus at the point

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