1. Being given the base and the vertical angle of a triangle, find the locus of the centre of its inscribed circle, and also of the centres of its exscribed circles.

2. Solve the equation

(x + a + b) (x + b + 2c) = (x + b − 3 a) (2 x + 2 b − 3a — 2c).

3. Find the simplest value of the fraction

A

5. In a plane triangle, prove the formula for tan in terms of the

2

sides, and write down the corresponding logarithmic equation.

6. Describe a circle passing through a given point and touching two given circles.

DR. TRAILL.

7. Draw a common tangent to two circles.

8. Prove that the bisectors of the angles at the extremities of the third diagonal of a quadrilateral inscribable in a circle are at right angles to each other.

II. Show how to calculate the distance between two places, inaccessible to an observer at either of two stations, the distance between which is known.

12. Prove that the area of the triangle whose sides are equal to the three bisectors of the sides of any triangle (abc) is equal to of the area of the triangle (abc) itself.

MR. PANTON.

13. If perpendiculars be let fall from any point in the circumference of a circle on the sides of any inscribed triangle, prove that the feet of these perpendiculars lie in a right line.

14. Inscribe a circle in a given sector of a circle.

15. If ABC be any triangle inscribed in a circle, and BD be drawn parallel to the tangent to this circle at A to meet AC (or AC produced) in D, prove that AB is a mean proportional between AC and AD. 16. Prove the formula

17. In any right-angled triangle, of which C is the right angle, prove the relation

1. In a given circle inscribe a triangle each of whose sides passes through a given point.

2. If a hexagon be circumscribed to a circle, prove that the three lines which join its opposite angular points pass through a common point.

5. Show how the sines and cosines of small angles can be calculated, and determine to six decimal places the values of sin 1° and cos Io.

7. A ship is approaching land, which is just visible from the masthead, and after a while land is just visible from the deck: prove that (x) the distance sailed over in the interval, is given by the equation

x = (√h - √h) √2r,

where h and h' are the heights of the mast-head and of the deck above the sea.

8. A mortgage is taken on an estate worth n acres of it; land rises p per cent. in price, and in consequence the mortgage is worth only n1 acres, and it is then paid off. During the continuance of high prices, another mortgage is taken worth n acres as before; but prices returning to their former level, the mortgage is worth n2 acres; prove the following relation :

and hence show that the advantage to the borrower from a rise in the price of land is less than the disadvantage to the lender from a fall in its price.

9. If any arc of a circle be divided equally and unequally, prove that the rectangle under the chords of the unequal parts, together with the square of the chord of the intermediate arc, is equal to the square of the chord of the half arc.

10. A purse contains ten coins, each of which is either a sovereign or a shilling: a coin is drawn, and found to be a sovereign, show that the chance that this is the only sovereign is

I

MR. PANTON.

11. If on any right line three pairs of points A, A'; B, B′; 0, C' be taken such that the rectangles OA.OA′, OB.OB', OC. OC' (0 being a fixed point on the line) are equal, then the anharmonic ratio of any four of the six points is equal to the anharmonic ratio of their four conjugates?

12. Prove that any four fixed tangents to a circle are cut by any fifth variable tangent in four points whose anharmonic ratio is constant.

13. Prove that a system of concentric circles is inverted from any point into a system of coaxal circles; and employ this theorem to derive properties of coaxal circles.

14. The centres of the exscribed circles of any triangle ABC are joined to form a triangle; if the radius of the circle inscribed in this triangle be denoted by r', prove the relation

r' (cos +cos B+ cos C) = r cot A cot B cot C,

where r is the radius of the inscribed circle of ABC.

15. Express cos" in terms of cosines of multiples of 0, distinguishing between the cases where n is even and odd.

4. Define a logarithm, and prove that log (an) = n log a for all real values of n whether integer or fractional.

5. Find the sum of an infinite number of terms of the series

I + 4x + 9x2 + 16x3 + 25xa + &c.,

where x is less than unity.

DR. TRAILL.

6. Prove that the present value of an annuity, to commence at the expiration of p years, and to continue for q years, is equal to the difference between its present value for p + q years, and its present value for

p years.

7. If

represent the probability of an event happening in one x + y trial, show that the probability of its happening at least t times in n trials is

and apply this theorem to determine the probability of throwing an ace twice at least, in three trials, with a single die.

8. Prove the theory of proportional parts in the use of logarithmic tables, viz., (1) that the increments of the logarithms of numbers are proportional to the increments of the numbers themselves; and (2) that the small changes in the logarithmic functions of an angle are nearly proportional to the increment of the angle.

9. Find the values of the following series :—

IÓ. Given three chords a, b, c of three arcs which together make up a semicircle, show that three circles can be described to which these chords are so related, and if d1, d2, dз be the three diameters of these circles, prove the equation

d1d2d3

= 2 abc.

MR. PANTON.

in a series of ascending powers of x, and find

12. If a, b, c be any three real numbers, not all equal, prove that the quantity

15. If four cards be drawn out of a pack, what is the chance that they

each belong to a different suit?