the relative economy of the section is expressed by the

1. Find the graphic sum, or line representative, of a

Р

series of n expressions of the kind x, where x is the

H'

abscissa from a given vertical axis of the line of action of any one of the n vertical and parallel forces P, and H a variable polar distance of given length for each force.

2. Prove the following rule for finding the central ellipse of inertia of the hollow triangle, whose centre of gravity is determined in Ex. 1, ch. iv. :—

From the angular points A, B, C draw lines through the centre of gravity O, meeting the opposite sides in D, L, E respectively. Upon DS, equal to 3DA, as diameter describe a circle, and at O erect a perpendicular to DA, meeting this circle in T; then OT will be equal to the radius of gyration as measured from O in the direction OA or OD. If, therefore, OT' and OT" be set off along OA and OD, lines drawn through T′ and T" parallel to BC will be tangents to the required ellipse. Repeat the same operation upon the lines BOL and COE, making the diameters of the circles equal to 3BL and CE, and drawing the tangents parallel to the opposite sides AC and AB respectively. Then complete the ellipse by § 11.

3. The angular points of the central kern of the hollow triangle (Ex. 2) lie in lines drawn parallel to the sides of the triangle at two-thirds of the altitudes of the triangle as measured from those sides.

4. Find the central ellipse and kern of a parallelogram

and shew that, if r be the radius of gyration measured from the centre of the parallelogram parallel to one pair of opposite sides and k the half height of the kern measured in the same direction, r2 = 1⁄2kh, h being the height of the parallelogram between the other pair of opposite sides; that the central kern is another parallelogram with its four apices at the ends of the central thirds of the secants through O parallel to the sides, and with its opposite sides parallel to the diagonals of the original parallelogram; and that any side of this kern parallel to one of the diagonals of the original parallelogram is harmonically separated from the nearer apex on the other diagonal by the two points in which the latter cuts the central ellipse.

5. The kern of a given plane triangle is a similar and, with respect to the centre of gravity, similarly placed triangle, whose dimensions are one-fourth those of the original triangle.

6. Find the central ellipse and kern of a trapezium ABCD, whose opposite sides AB and CD are parallel; and shew that, if a semicircle be described on the perpendicular distance GH between the parallel sides AB (2a) and CD (26); if, then, from the intersection I of the diagonals AC and BD, a perpendicular be erected cutting off the half chord XY in the semicircle; and lastly, if from the summit K of the semicircle the distance KL, equal to XY, be set off along KG, the line LH will be thrice the radius of gyration of the central ellipse of the trapezium relatively to the diameter drawn parallel to the parallel sides AB and CD; whilst the half-diameter, parallel to AB, of the same ellipse is equal to √1⁄2(a2 + b2).

2

7. Verify the central ellipses and kerns of the rail and angle-sections represented in Figs. 91 and 92.

8. If a series of chords be drawn through a point V within a parabola and tangents be drawn at the points of contact on each chord, such pairs of tangents will form

a series in involution and will determine, by their intersection with any tangent to the curve, an involute range. Find the centre and constant of the involution (§§ 11, 15, 27).

9. As a particular case, shew that (1°) the tangent at the vertex of a parabola is divided by the pairs of tangents at the ends of focal chords into an involute range, (2°) that any other tangent of the same group is also divided in involution by the remaining tangents. Find the centre and constant of involution in each case.

10. If from any centre three pairs of rays be drawn parallel respectively to the three pairs of opposite sides of a complete quadrangle, they will form three pairs of rays of an involute pencil. If two of the three pairs of opposite sides meet at right angles, the third pair will also meet at right angles; or the three perpendiculars let fall from the three apices of a triangle upon the opposite sides meet in a point, hereafter termed the centre of altitude of the triangle ($ 25).

11. The sides of a triangle and the line at infinity in the same plane form a complete quadrilateral whose three pairs of opposite apices are projected from the centre of altitude of the triangle by three pairs of normal rays in involution (§ 25).

12. If a parabola be enveloped by a complete quadrilateral whose fourth side is the tangent at infinity, the centre of the rectangular involute pencil projecting the three pairs of opposite apices of the quadrilateral will coincide with the centre of altitude of the triangle formed by the first three sides (Ex. 11).

13. The tangents at the ends of a focal chord meet at right angles in the directrix (§§ 30, 36).

14. The locus of the centres of altitude of the triangles formed by the tangents of a given parabola, taken three at a time, is a straight line (§ 21 and Ex. 13).

15. The systems of points A, B, C and A', B', C' on a conic are projectively allied; find their double points (§ 37).

16. The systems of points A, B and A', B' on a conic are projectively allied in involution; find their double points (§ 21).

17. Compare the moduli of the sections given in Figs. 86, 91, and 92, and class them in the order of economy.

CHAPTER VI

THE ELASTIC LINE

65. DEFLECTION.-If M represent the bending-moment, E the coefficient of elasticity, and I the moment of inertia of a cross-section of a beam; the angular deflection between two sections of the beam, whose distance apart is dx, will be

Now if M,, M2, M3, etc., be the bending-moments at successive cross-sections of the beam, whose common distance apart is dx; and if, with the series of values M1dx, M ̧dx, M.dx, etc., treated as a series of forces, we set off the line

of forces (Fig. 93), and then draw the funicular AB (Fig. 94), relatively to the pole O and the polar distance EI; the