according as c is greater or less than a.

=

Hence there are

always, except when a 2c, four real points at which the curve crosses its oblique asymptotes.]

To explain how the form of the curve changes gradually from fig. (16) to fig. (18) as a passes through the value C, we must observe that, when a = c, the straight line AE is part of the locus; and that the curved branch cuts AE in F, AF being equal to 2a: also that when a = c, A and C coincide. It appears then, that as A approaches C, the arc CG becomes less curved, and approximates to the straight line AF. Similarly the branch HK becomes less and less curved, and at last coincides with FE. Also as H and G approach F, the two branches LH and GM ultimately unite and the curve assumes the form of fig. (17). It is clear that the curvature at G and at H must increase indefinitely as the curve fig. (16) approaches its limiting form. The above explanation holds for the change from fig. (18) to fig. (17).

9. SPHQ is a quadrilateral, P and Q being points in an ellipse of which S and H are the foci; if Q be fixed while P moves, find the locus of the centre of gravity of the perimeter of the quadrilateral.

Let G1, fig. (19), be the centre of gravity of SP and PH,

SQ and QH;

G2 then G the centre of gravity of the whole perimeter is the middle point of GG2, and since G2 is fixed the locus of G will be similar to the locus of G1, and of half the linear magnitude: also when PCQ is a straight line, G will be at C.

1

To find the locus of G. Join UV the middle points of SP, PH: UV evidently passes through G. Again, a perpendicular CP' from C upon the tangent at P also passes through G1; for if SP be produced to H', so that PH' = PH, CP', which is parallel to HH' and bisects SH, will also bisect SH', (Euc. VI. 2); therefore a line SH' would balance on CP'; and since PH and PH' are equally inclined to CP', SP and

PH will balance on CP'. coordinates of P and G1,

y

Draw GN; then if (xy), (x,y,) be

y1 = 1⁄2), x1 = y1 tanCG ̧N=y, tan (inclination of tangent at P to axis of x)

2

the equation to an ellipse whose semi-axes are half the semi-latus-rectum, and half the minor axis of the given ellipse; therefore the locus of G is an ellipse whose semi-axes are onefourth of the semi-axis minor, and one-fourth of the semi-latusrectum of the given ellipse, passing through C, having its centre on the perpendicular from C upon the tangent at Q, and its major axis perpendicular to the major axis of the given ellipse. HP = r' ;

SP = r,

Otherwise. Let and let (xy), (x'y'), (xy) be coordinates of P, Q, G respectively.

10. From an external point P two tangents are drawn to

y2

an ellipse + b2

= 1. Supposing the locus of the centre of

gravity of the triangle, included between the two tangents

and the chord of contact, to be an ellipse

the equation to the locus of P.

y2
+ =

2

1, find

What must be the relation between a, b, a1, b1, in order that the locus of P may be an ellipse?

Let h, k, be the coordinates of P, and (x,, y1), (x, y), be the two points of contact. The equation to the chord of contact is

Now, x, y, denoting the coordinates of the centre of gravity of the triangle,

This equation cannot be reduced to one of the second order unless

a': b': a:b;

under this condition it will plainly represent an ellipse, its equation being of the form

h2

k2
+ = = n2.
a2
α b

b'b:

COR. Let a = a, b' b: then the equation to the locus of P becomes

(+) {

The former result shews that the locus of P is an ellipse with axes 4a, 46. The latter belongs to the case when the triangle is constantly zero.

11. The radii vectores of any series of points in the path of a particle, moving about a centre of force, being in arithmetical progression, the times of arriving at these points, reckoned from a given epoch, form another arithmetical progression. Find the equation to the path.

By the condition of the problem, it is plain that dt is constant when dr is constant; but, t being some function of r which we may denote by f(r),

12. In any machine in which two weights P and W are suspended by strings and balance each other in all positions, let P be replaced by a weight Q equal to pP; if in the ensuing motion W and Q move vertically, find the tensions of these strings, neglecting the friction of the machine and the masses of its several parts.

Let WmP, then, by the principle of virtual velocities, P describes a space m times as great as W in the same time; and after P is replaced by Q, Q must describe m times the space described by W in the same time; therefore the whole accelerating force on Q must be m times as great as that on W. Let T, T" be the tensions of the strings to which Q and W are attached, then

and since the machine has no inertia, the forces which act on it must have the same relation as if it were at rest, (otherwise a finite velocity would be instantaneously generated,) therefore

T" T