THURSDAY, Jan. 19, 1854. 9...12.

1. Two circles of radii r, r', touch a straight line at the same point on opposite sides: a circle, of which the radius is R and of which the straight line is a chord, touches both the former circles. Prove that the length of the chord is equal to

4R

(5)+(笑)

Let AB (fig. 38) be the straight line, E the point in which it is touched by the two circles, the centres of which are O, O'. Let C be the centre of the third circle. Draw CH at right angles to AB. Join 00', OC, O'C.

Let CH=α, HE=b, LOCH=0.

From the geometry,

Similarly, putting -a for α, and r' for r,

(3).

(4).

2. Prove that, n being any positive integer, and e the base of Napier's logarithms,

(n+1)"

en

LEMMA. For any value of x, except zero, between the limits

and therefore y keeps always decreasing. Again, as x increases

dy

from 0 to ∞, ax

is always positive, and therefore y keeps always increasing. But y=0 when x=0: hence the truth of the lemma.

Since, when x is any positive quantity,

Writing for n, successively, 1, 2, 3, ... n, we have

Multiplying these inequalities together and casting out factors common to both sides of the resulting equation, we have

(n + 1)"

e". 1.2.3 ... n > (n+1)”, or e"> 1.2.3 ... N

3. From a focus S of a conic section ARQPA (fig. 39), three radii vectores SR, SQ, SP, are drawn, the angles PSQ, QSR, being invariable. Prove that the tangent at P intersects the chord RQ produced in a point of which the locus is another conic section.

Supposing e to be the eccentricity of the original conic section and e' of the conical locus, shew that, if RSQ = 2a, and QSP B,

=

+

Let ASQλ. Then the equation to the chord RQT is

At the intersection of the chord and tangent, subtracting and adding the equations

4. Tangents PP, PP", are drawn from a point P to touch the ellipse

a2

x2
+

y2

2

1,

at points P', P". Supposing the harmonic mean between the abscissæ of the points P', P", to be equal to that between their ordinates, shew that the locus of P consists of four arcs of a curve of the third order.

Trace the curve and shew that, when a = b, the curve degenerates into a straight line and an ellipse.

Let h, k, be the coordinates of P; x, y, of P'; x, y, of P". The equation to P'P" is

At the intersections of this line with the ellipse,

or, replacing h, k, by x, y, we have for the equation to the curve in which P always lies,

The shape of the curve is IEBA'OABE'I', (fig. 40), IOI' being an asymptote.

The curve at O is inclined to the axis of x at an angle

The locus of P consists of the four arcs

IE, BA', AB', E'T'.

At the intersections of the ellipse and curve

If a = b, then the equation (1) becomes

(x − y). (x2 + xy + y2 — a2) = 0,

which represents a straight line EE" and an ellipse AaB'B' A'a' Bẞ,

(fig. 41), the semi-axes of which are

The locus of P consists of the lines

EF, E'F", Ba'A', B'aA.