Let OFGE be the cutting plane;

a, ẞ the angles AOE, BOF respectively.

Describe a spherical surface with O as centre, meeting the planes AC, BC, FE in the arcs hl, lk, kh respectively: then hl= a, lk = B, hlk; and by the given condition the arc drawn from 7 perpendicular to hk = 1; therefore

1=

2

= tan2π cot2 a + tanπ cot2ß = cota + cot2 B = p2 + q2...(1), writing p and q for cota and cotß.

Again, the equation to the cutting plane referred to OA, OB, OC as axes, is

therefore the volume of the part between OBDA and OFGE

And since this is always the smaller part, the two will be most nearly equal when this part is the greatest possible;

therefore the plane COG is perpendicular to the plane FE, which defines the position of the cutting plane.

7. If r, r', be the radii of curvature of an involute and evolute at corresponding points (x, y), (x', y'); prove that

2

=

ellipse of which the semiis equal to

dx d'y - dy d3x

(dx'd3y' — dy'd3x')' _ (dx d'y - dy d'x)" dy

(dx2 + dy'2) 3

1 (dx d'y - dy d2x)*

=

=

dx2+ dy dy'* dx2 dx" + dy'

dx

(1),

r

3

a2b2

xy.

Now xy is a maximum, under the condition +

has a maximum value equal to

(응-음).

2 b

first supposing a to be less than c, then equal, then greater; and shew how the three forms of the curve pass into each other, when the value of a is supposed to increase gradually through the value c.

To find the asymptotes.

x = O and x = a each make y = ∞ ;

a

therefore y = ± x + is a pair of oblique asymptotes, and

2

if we consider points far enough from the origin, the asymptotes lie between the curve and the axis of x.

therefore the form of the curve is that given in fig. (16), where OA a, OC = c, OD = c, OB = a.

=

or rather we should say that when xc, y may have any value. Thus the line x

c = 0 is part of the locus.

To find the general form of the rest of the locus,

*The notation in the text is used for stating concisely whether the value of y is possible or impossible between particular values.

The form of the curve is given in fig. (17), where OA = c, OD = c, OB = c.

The form of the curve is given in fig. (18), where OA = a, ов =

a, OC OD :

=

= C.

[It will be easier to draw the curve if we find the points. where it cuts its oblique asymptotes. The abscissæ of these points are given by the equation

c, except when a =

and it may be shewn that the negative value is always less than 2c, in which case the value of x isc also that the positive value is greater or less than c