13. There are generally two directions in which a projectile may be projected with given velocity from a point A, so as to pass through another point B; shew that one of these directions is inclined to the vertical at the same angle that the other is inclined to the line AB. Hence shew that the locus of points, for which a given sight must be used in firing with a given charge of powder, is the surface generated by the revolution, about the vertical, of the path of the bullet obtained by aiming at the zenith with the given sight, and with the given charge of powder.

The former part of this problem is solved in Phear's Dynamics, Sect. III. Art. 30, and the latter part follows at once.

Or the latter part may be worked independently as follows: To find the locus of points, for which the same sight must be used.

Let a be the inclination of the line of the sights to the axis of the barrel; r, the polar coordinates of a point for which this sight is adjusted; then, substituting r cose and r sin for x and y in the ordinary equation to the path of a projectile, a being the angle of projection,

r{sino cos(0+a) — cose.sin(@+a)} cos(0 + a)

272 (r cose)2 2√2

sinė sina) = ( cose)2

9 2√2

2 V2 sin'a

the equation to the path of a particle projected with velocity V at an angle a to the vertical: that is, if a man, facing the south for instance, aim, with a given sight, at the zenith, the ball, which falls behind him, will pass through all those points to the north of the man, for which the given sight is adapted.

14. A prism whose base is a given regular polygon is surmounted by a regular pyramid whose base coincides with the head of the prism; find the inclination of the faces of the pyramid to its axis in order that the whole solid may contain a given volume with the least possible surface.

Let a be the perpendicular distance of one of the sides of the polygon from its centre; the inclination of a face of the pyramid to the axis; x the height of the prism; A the area of polygon; P the perimeter. Then

15. An ellipsoid is intersected in the same curve by a variable sphere, and a variable cylinder: the cylinder is always parallel to the least axis of the ellipsoid, and the centre of the sphere is always at one focus of a principal section containing this axis. Prove that the axis of the cylinder is invariable in position, and that the area of its transverse section varies as the surface of the sphere.

Let e, &, be the eccentricities of the two principal sections through c.

y2

(x − ae)2 + y2 + z2 =

b2

c2

.......ellipsoid,

the equation to that cylinder which intersects the ellipsoid in the same curve as the cylinder.

Therefore the axes of all the cylinders coincide with a directrix of the principal section, and the area of a transverse section varies as 8" and therefore as the surface of the sphere.

16. An elastic tube of circular bore is placed within a rigid tube of square bore which it exactly fits in its unstretched state, the tubes being of indefinite length; if there be no air between the tubes, and air of any pressure be forced into the elastic tube, shew that this pressure is proportional to the ratio of the part of the elastic tube that is in contact with the rigid tube, to the part that is curved.

Let ABCD (fig. 20) be a section of the rigid tube, EGHF part of the section of the elastic tube: it is clear from symmetry that if E and F be the middle points of the sides AB, AD, the part EGHF is one-fourth of the perimeter of the elastic tube. Also the free portion GH is circular: for the pressure and tension being the same at every point, the radius also must be the same, by the formula T= pr. Also, since the pressure is finite the curvature must be finite throughout, so that the sides of the rigid tube, with which the elastic tube coincides for a certain space, must be tangents to the free portions of the elastic tube: the circular are GH is therefore a quadrant.

2a, EOG = 0, p

= the pressure of the air within

Join OG, and draw the radii GK, HKQ.

Let AB

the tube.

=

Consider an annulus of the elastic tube whose breadth is the unit of length; and let T be the tension of this portion, E being the tension required to stretch this annulus to twice its natural length. Then

therefore the pressure of the air in the tube is proportional to the ratio of the part that is in contact to the part that is curved.

17. OA, OB, are any equal arcs of two given great circles of a sphere, intersecting in 0. A and B are joined by an arc of a great circle, and also by an arc of a small one described about 0. Find the area of the lune included between the two joining arcs.

If OA=λ and LAOB=2w, prove that the lune is greatest

ACB (fig. 21) is the arc of the small circle,

AC'B is the arc of the great circle:

d'u

Since is negative, this result corresponds to a maximum

value of u.

18. The ridges of two roofs are at right angles to each other, and the inclination of each roof to the horizon is 0; the shadow of a chimney falling upon them makes angles a and with their ridges; shew that

Let ACDB (fig. 22) be one side of the shadow on one roof; through C draw the vertical CE, and through D draw a horizontal plane cutting CE in E, and meeting the roof in DF, which is parallel to the ridge; draw EF perpendicular to DF, and join CF, DE.

Now CFE

= inclination of roof to horizon =

0, CDF = inclination of shadow to ridge = a;

and since CDE is the vertical plane passing through the sun,