EDF is equal to the sun's azimuth measured from the direction of the ridge of the roof, suppose. Then = 19. The hour angles of two stars being e, e', and the azimuths a, a', when a a' has for a moment a stationary value; prove that the latitude λ of the place of observation is given by the formula Differentiating these two equations, a, e, a', e', being variables, cosec a. sine. (cota' cose' + sinλ sine') =cosec'a'. sine'. (cota cosa + sinλ sine), sine. sine'. sinλ (cot'a — cot'a') = sine'. cose. cota. cosec a' sine. cose'. cota'. coseca, sine. sine'. sinλ (sin a' cos'a - sina costa') =sine'. cose. sin 2a 2 sinλ. sin (a' — a) . sin (a' + a) = sine. cosɛ'. sin 2a', sin 2a. cotesin 2a'. cots', sin 2a. cote' cos 2a cos 2a 20. A thin hollow ring, of which the plane is vertical, and which contains a bead, is placed upon a smooth horizontal plane: prove that the bead, having been placed near the lowest point of the ring, will oscillate isochronously with a perfect pendulum the length of which is equal to a being the radius of the ring, μ its mass, and m the mass of the bead. Let C (fig. 23) be the centre of the ring, A its point of contact with the horizontal plane, Ox the rectilinear locus of A, O being a fixed point. From P, the place of the bead, draw PM at right angles to Ox. Let OA = x, OM = x', PM = y', LACP= 0, R = the mutual action between the ring and the bead. x = x + a sin, y' =α a cose. As far as the first order of small quantities, 21. A uniform rod, not acted on by any forces, is in motion, its ends being constrained to slide along two fixed rods at right angles to each other in one plane. Prove that, during the whole motion, the wrenching force at any point of the moving rod varies as the product of the distances of the point from the two fixed rods. Let AB (fig. 24) be the moving rod, O being the intersection of the two fixed rods. Let C be any point in AB. Draw CH, CK, at right angles to OA, OB. Let AB = 2a, AC=2u, BC= 2v, LBAO = 0, m = the mass of AB, w = its angular velocity, which will be invariable. The actions and reactions and the wrenching force are indicated in the figure. ти (2a-u) w3 cos 0 = X .....(1). α mu и a = = = mu Xu sin0 +2 Yu cos 0 a w2 sine cose w'u sine cose {u (2a — u) — 2 (a3 — u3) — u*} mw. 2u sine. (2a - 2u) cos 0 ∞ CH.CK WEDNESDAY, Jan. 18, 1854. 9...12. 1. THERE are n points in space, of which p are in one plane, and there is no other plane which contains more than three of them; how many planes are there, each of which contains three of the points? Conceive n points such that no plane contains more than three of them; the number of planes, each of which contains three points, being equal to the number of combinations of n things taken three at a time, is equal to now if p of these n points be selected, the number of planes, each of which contains three of these points, is hence, if these p points move so as to lie in one plane, this 2. A bag contains nine coins, five are sovereigns, the other four are equal to each other in value; find what this value must be, in order that the expectation of receiving two coins at random out of the bag may be worth twenty-four shillings. |