6. Shew how to find the coordinates of the centre of pressure of a plane area immersed in homogeneous liquid. If a straight line be taken, in the plane of the area, parallel to the surface of the liquid and as far below the centre of inertia of the area as the surface of the liquid is above, the pole of this straight line with respect to the momental ellipse at the centre of inertia whose semi-axes are equal to the principal radii of gyration at that point will be the centre of pressure of the area. Taking the principal axes at the centre of inertia as coordinate axes, if the equation to the momental ellipse be x2 y2 ffxdxdy Sfy dx dy + then a2 = b2 = ffdx dy Jdx dy If x cosa + y sin a=p be the equation of the line in the surface of the liquid and (xy) the centre of pressure, Therefore (xy) is the pole of x cosa+y sina = -p with respect to the momental ellipse. vii. Shew that a cloud of small particles or of fine dust, if only deep enough, however far the particles may be separated in comparison with their diameters, can give a brightness equal to half that of a slab of the same material similarly illuminated by a distant source of light. Hence shew that the brightness of a comet, and the visibility of a star through the head of a comet, are consistent with the comet's being a mere swarm of meteorites. Shew how to compare, approximately, the utmost brightness of a cloud nearly opposite to the sun, and consisting of small spheres of water, with the brightness of the sun's image in a pool. Suppose the cloud arranged in layers, parallel or not, each allowing 1-e of the incident light to pass, and sending back ef of it. Then 1st layer gets 1 sends back ef, 2nd............ 1 -e ....... ..... (1-e)'ef, 3rd............ (1-e) .............. (1 − e)‘ef, the factors (1-e), (1 − e)", ...., being squared, because of the loss in coming out. Therefore, altogether there is sent back when e is small. But, if e=1, no light gets through the first layer, which is then virtually a slab; and it sends back f, double the light of the cloud. Comets (except occasionally their nuclei, which are, probably, to a great extent self-luminous) are not nearly of half the brightness of planets equally distant from the sun. Through the cloud, above spoken of, the brightness of a star would be reduced from 1 to (1-e)", where n is the number of layers. The individual particles of the comet are usually too far apart to eclipse more than a small fraction of the disc of a fixed star, even though the disc is invisible in our best telescopes. For the last rider using the same process, and taking account of the size of the images in the reflecting spheres; if 2w be the angular diameter of the sun, and r the radius of a raindrop, the area of the image of the sun formed by reflection at the convex surface of the raindrop will be 17" sin2w, and therefore the ratio of the apparent size of the image of the sun to the apparent size of the raindrop will be sin3w. Therefore the cloud will give at most a brightness which will bear to the brightness of the image of the sun in a pool the ratio sin'w, which is the ratio of the apparent area of the sun to the apparent area of the hemispherical celestial vault. The light reflected from the interior of the rain-drops, by which the rainbows are formed, is insensible nearly opposite the sun. N viii. If 0, 4, be the angles of incidence and emergence of two parallel rays passing through a prism in a plane perpendicular to the edge; d, d, the distances between these rays before incidence and after emergence; shew that where 80 is any small change of 0, and Sp the corresponding change of 4. Shew from this that the position of minimum deviation. is that of most distinct vision through a thin prism. If d be the distance between the rays inside the prism, In the position of minimum deviation dd, and therefore the divergence of the rays is unaltered by the prism, which is the condition for the most distinct vision. ix. Discuss separately, and without formulæ, the effects of annual parallax and aberration on the apparent position of a fixed star as it would be seen from a comet of a year period, moving in the ecliptic, in a path of great excentricity. Compare these with the corresponding effects as seen from the earth. Trace the curves representing, from each of these points of view, the apparent annual path of a star, without proper motion, situated near the pole of the ecliptic. The effect of annual parallax is to make the star describe an equal and parallel orbit turned through 180°, and the projection of this on the celestial sphere will be the apparent path of the star due to annual parallax. The effect of aberration is to make the star describe the hodograph of the orbit, which is a circle parallel to the plane of the orbit, and the projection of this on the celestial sphere baca be the apparent path of the star due to aberration. If the star be near the pole of the ecliptic, the orbits due to annual parallax and aberration will be unprojected on the celestial sphere. The orbits seen from the comet will be an ellipse of great excentricity, and a circle passing very nearly through the mean position of the star (figs. 55, 56, 57) and the apparent annual path will be the resultant of these two orbits; seen from the earth the orbits will be approximately concentric circles. 10. Explain the construction of charts on the gnomonic, the stereographic and Mercator's projections. Examine what curve in each case will represent (a) a rhumb line, (B) a great circle. Shew how to draw the trace in the two first projections of the great circle passing through any two given places. Prove that the equation of the trace on a Mercator's chart of a great circle will be always of the form 2 sin (~~+a) = 7 (că — e ̄3), α where a is the radius of the sphere. The traces of (a) a rhumb-line are respectively a transcendental spiral of the form 2a Ө = e-e, an equiangular spiral and a straight line; and the traces of (B) a great circle are respectively a straight line, a circle, and the transcendental curve of the third part of the question. To draw the trace of a great circle joining two given points in the gnomonic projection, draw the straight line joining the projections of the points; in the stereographic, draw the circle passing through the projections of the points and their antipodes. Let PM be the great circle (fig. 58), OX, OY the equator and initial meridian which project into the coordinate axes in Mercator's projection, and let 7, λ be the longitude and latitude of P, and x, y its coordinates on the chart. |