If the internal bisectors p, q, r of the angles of a spherical triangle make angles a, 6, y, with the opposite sides prove that

The Board of Examiners.

1. Find the general equation of a straight line through the intersection of the lines

Find the equation of the straight line through this point and through the point of intersection of а1x + b1y + c1 = 0

a1x + by + c1 = 0.

2. Find the equation of the chord of contact of tangents drawn from the point x'y' to the circle.

x2 + y2 = a2.

Two triangles are such that the angular points of one are the poles of the sides of the other with respect to a circle. Shew that the lines joining corresponding vertices meet in a point.

3. Show that three normals can be drawn from a given point to a parabola.

If the line joining the feet of two of the normals pass through a fixed point, then the point from which the normals are drawn traces out a parabola.

4. Find the equation of the chord joining two points on an ellipse.

Two tangents are drawn to an ellipse. If the chord of contact subtends a right angle at the centre shew that the chord of contact touches a concentric circle, and that the intersection of the tangents traces out a concentric ellipse.

5. Find the equation of a hyperbola referred to its asymptotes as axes.

If a variable point on a hyperbola be joined to two fixed points on the curve the joining lines intercept a fixed length on each asymptote.

7. State and prove Lagrange's formula for the remainder in Taylor's series.

If u be a homogeneous function of n dimensions in any number of variables, x, y, z, &c., prove that

Shew how to extend this result to differential coefficients of the second and higher orders.

9. State and prove a rule for finding maximum and minimum values of a function of one independent variable.

Find the maximum and minimum values of

(x − a)a (x − b)13(x − c).

11. Shew how to find the partial fractions corresponding to a repeated quadratic factor in the decomposition of a rational fraction.

The Board of Examiners.

1. Through five points, no four of which are in a straight line, one conic and only one conic can be drawn.

Five fixed points are taken, no three of which are in one straight line, and five conics are described each bisecting all the lines joining four of the fixed points, two and two; prove that these conics will have one common point.

2. The locus of the pole of a given straight line with respect to a series of confocal conics is a straight line.

Two tangents OP, OQ, being drawn to a given conic, prove that two other conics can be drawn confocal with the given conic, and having for their polars of O the normals at P, Q.

3. Find the envelope of the straight line

when

lx + my + nz = 0

al2 + bm2 + cn2 + 2fmn + 2gnl + 2hlm = 0.

A triangle is inscribed in an ellipse and two of its sides pass through fixed points; show that the envelope of the third side is a conic having double contact with the ellipse.

4. The polars of a fixed point with respect to a system of conics passing through four fixed points will pass through a second fixed point.

If the first fixed point lie on a given straight line then the second fixed point will lie on a conic.

Hence or otherwise shew that the locus of the pole of a given straight line with respect to the conics which pass through four given points is a

conic.

5. Shew that any quadrilateral may be projected into

a square.

Prove that the eight points of contact of two conics with their common tangents lie on a conic.