ray towards the refracting edge of the first prism will be i+24-i-24-i-i. But if the prisms be related as in fig. 24, will be the angle of incidence on the second prism and the total deviation of the ray will be i+2p+-24 = i + i'. x. Define the illumination at any point of a surface and prove that the illumination due to rays proceeding from a bright point varies directly as the cosine of the angle of incidence and inversely as the square of the distance from the bright point. Explain how units of brightness and illumination could be selected and defined. The unit of brightness is defined by a light which consumes a certain amount of oil or gas in the unit of time; for instance, the light of a wax candle of certain weight to burn a certain time. The unit illumination would be that produced at unit. distance from the light of unit brightness. xi. Shew how to find the focal length of a system of lenses of known focal lengths whose axes are coincident and which are separated by given intervals. If the lenses be all concave, each of focal length f, and such that the interval between the rth and (r+1)th lenses is equal to the distance of the focus after the rth refraction from the rth lens, and if the original pencil be parallel, prove that the distance of the nth focus from the nth lens is ... 27-1 f. 2′′ – 1ƒ. Let V31 19 respective lenses, then F xii. Describe the Newtonian telescope and find an expression for the magnifying power. If the focal length of the reflector be 2 feet and the focal length of the eye-glass 1 inch, and if the instrument be in focus for a star to a person who sees most distinctly at a distance of 6 feet, prove that it requires no readjustment for a person who sees most distinctly at a distance of 2 feet and is viewing an object whose distance is 609 yards. Let F, F' (fig. 25) be the images of the star and object formed by the first mirror; f, f' the images formed by the small mirror; g, g' the images formed by the eye-piece. 1. UPON the sides of a triangle ABC as bases are described three equilateral triangles aBC, bCA, and cAB, all upon the same side of their bases as the triangle ABC. Prove that Aa, Bb, Cc are all equal and pass through a point which lies on all the three circles circumscribing the equilateral triangles. If Aa be produced (fig. 26) to meet the circle described round aBC in P, then aPC=120° and ▲ aPB= 60°; therefore BRC= 60°. = Since APC 120°, therefore the circle described round bCA will pass through P, and bPC=2bAC=60°. Therefore BbP is a straight line. = Since APB 60°; therefore the circle described round CAB will pass through P, and cPB=LcAB=60°. Therefore cCP is a straight line. In the triangles AaC and BbC, a C=BC, CA=Cb and L AaC=LbBC; therefore Aa = Bb. In the triangles BaA and BCc, Ba= BC, BA= Bc and L BaA=LBCc; therefore Aa =Cc. ii. Given the circumscribed and inscribed circles of a triangle, prove that the centres of the escribed circles lie on a fixed circle. The circle circumscribing the triangle ABC is the ninepointic circle of the triangle A'B'C' formed by the centres of the escribed circles. If D be the centre of the inscribed circle, and O of the circumscribed circle of the triangle ABC, then A', B', C' lie on a circle of radius double the radius of the circumscribed circle and with centre at O' on DO produced, such that DO' = 2DO. 3. Out of m persons who are sitting in a circle three are selected at random; prove that the chance that no two of those selected are sitting next one another is If A be the first person selected, the chance B is not next 2 If A and B be next but one, the chance of which is C must not be next either, the chance of which is m 2 If A and B have at least two people between them, the 4. A person has n sewing-machines, each of which requires one worker and will yield each day it is at work 9 times the worker's wages as nett profits; the machines are never all in working order at once, and at any time it is equally likely that 1, 2, 3 or any other number of them are out of repair. The worker's wages must be paid whether there is a machine for him to work or not. Prove that the most profitable number of workers to be permanently engaged will be the integer nearest to ng 2+1 1 |