tables, to 3 places of decimals, the number of which 1.3 is

20. Explain what is meant by systems of logs calculated to different bases; what is the base of each of the systems most commonly used in mathematical calculations? State briefly any advantage belonging to these systems. Show log,N=3 log,N.

21. If all the logs in the common tables were doubled, they would still be the logs of the same numbers as before, but the base would now be 10. Prove this. What would the bases be if the existing logs were (a) multiplied by n, (B) divided by n?

m

22. Find log a to the base an.

23. Show that the characteristic of the log of a number can be determined by inspection when the base is 10. From log102 and log107, find log10'0000432.

24. Explain why log103-56 and log10 0356 have the same mantissa log10 0356 and log10 04 have the same characteristic.

Show that

1 1
log Palog P; log.P=

n

log P

log.a

25. Assuming that

log.(1+x)=x—x2+}x3 —‡x1+&c., show that log.(n+1)

2 1

2

1

2

=log.n+2n+1+3* (2n+1)3+5* (2n+1)3 ́

+ &c.

26. Investigate the expansion for log (1+x) in powers of x, and show that the series is convergent when x is less than unity.

27. If e be the base of Napier's system of logarithms, expand e, and show that the series for calculating e converges.

28. Investigate an expression for the logarithm of a number to a given base, in the form of a converging series. 29. Obtain the series

1. AB is the side of an equilateral triangle inscribed in a circle whose diameter is AOC and centre O.

Prove (1) the triangle BOC is equilateral;

(2) AB2=3 AO2.

2. If two chords AB, CD in a circle cut each other at right angles, the sum of the opposite arcs AC, BD will be a semicircle.

3. The angle between the lines bisecting two adjacent exterior angles of any quadrilateral is equal to the angle between the lines bisecting the two interior opposite angles.

4. If through the extremities of a chord common to any number of circles two lines be drawn cutting the circles, the lines joining their other points of intersection with each circle are parallel.

5. In a triangle the sides about the vertical angle are 25 and 16, the line bisecting the vertical angle is 12. Find the base.

6. If AB, CD be the opposite sides of a quadrilateral circumscribing a circle whose centre is E, prove that the angles AEB and CED are together equal to two right angles.

7. The sides AB, AC of a triangle are bisected in E, F. AD is drawn perpendicular to the base BC. If ED, DF be joined, the angle FDE is equal to the angle BAC, and the area of the triangle ABC is double that of the quadrilateral AFDE.

8. Two circles touch each other; a common tangent is drawn to them at their point of contact; from any point in this tangent a tangent is drawn to each circle. Show that these latter tangents are equal to each other.

9. Through any point D in the base of a triangle ABC, straight lines DE, DF are drawn parallel to the sides AB, AC, and meeting the sides in E, F, and EF is joined. Prove that the triangle AEF is a mean proportional between the triangles FBD, EDC.

10. Given two straight lines, construct the rhombus of which they are the diagonals, and show that it is greater than any other parallelogram having the same straight lines for diagonals.

11. Prove that if a quadrilateral be bisected by both of its diagonals, it must be a parallelogram.

12. Prove that the area of the isosceles triangle is greater than that of any other triangle on the same base, and having an equal perimeter.

13. ABC is a triangle inscribed in a circle, BD is drawn

parallel to the tangent to the circle at the point A, and meets AC in D. Prove AC: AB:: CB: DB.

14. From any point in the base of a triangle lines are drawn parallel to the sides. Show that the intersection of the diagonals of every parallelogram thus formed lies in a certain straight line.

15. If two sides AD, BC of a quadrilateral inscribed in a circle ABCD be produced to meet in E, the circle described about ECD will have the tangent at E parallel to AB.

16. If from the angles at the base of any triangle perpendiculars be drawn to the opposite sides, produced if necessary, the line joining the points of intersection will be bisected by the perpendicular drawn to it from the middle of the base.

17. ABC is a triangle, C an obtuse angle, BF the perpendicular on AC produced. If in the base AC a point E be taken, such that AC=2 EF, prove AB2 + BC2= 2 (AE2+ BE2).

18. Prove that the rectangle contained by a side of the isosceles triangle, Euc. IV. 10, and the base, together with the square on the base, is equal to the square on a side of the triangle.

19. In an isosceles triangle show that the base is a mean proportional between the diameter of the circle inscribed in the triangle and the diameter of the circle that is escribed touching the base, and the equal sides produced.

20. The sides of any quadrilateral figure are together greater than the sum of its diagonals.

21. The sides AB, AC of a triangle are bisected in D and E, and BE, CD are produced until EF=EB and GD=DC. Show that the line GF passes through A.

22. Perpendiculars are let fall from two opposite angles of a rectangle upon a diagonal. Show that they will divide the diagonal into equal parts if the square on one side of the rectangle be double that on the other.

23. Describe a circle touching one side of a triangle and the other two sides produced.

24. Given the centres of the three escribed circles of a triangle; construct the triangle.

25. If E be a point without a circle, and straight lines EAB, ECD are drawn to cut the circle in A, B and C, D, show that the angle CEA is equal to half the difference of the angles subtended by AC and BD at the centre of the circle.

26. Show that the square on the sum of two lines, together with the square on their difference, is double the squares on the two lines.

27. If two circles intersect in A and B, and through P, any point in the circumference of one of them, the chords PA, PB be drawn to cut the other circle in C and D, show that CD is parallel to the tangent at P.

28. The sum of the alternate angles of any hexagon inscribed in a circle is equal to four right angles.

29. ABC is an isosceles triangle, C being an obtuse angle, AD a perpendicular drawn from A to BC produced; the square on AB is equal to twice the rectangle contained by BC and BD.

30. If two circles intersect in A and B, and AC, AD be two diameters, prove that the line CD will pass through B.

31. If two circles cut each other, show that there is a certain straight line from any point of which exterior to the circles the tangents drawn to both circles will be equal.

32. A pyramid is cut by a plane parallel to the base. Show that the section is a figure which is similar to the base.

33. P is a point without any number of concentric circles, O is their common centre; from P any number of tangents are drawn to the circles. Show that the circle described on PO as a diameter will pass through all the points of contact.

34. If the straight line AB be divided in C, so that the rectangle contained by AB and BC is equal to the square