tively, having their centres in the axis of the cylinder; shew that the distance of these surfaces, in order that an eye placed at the concave surface may see the image of a distant object distinctly, must = u (r - 8 ; and that the magnifying power п M 1 will S 152. A small pencil of diverging rays, whose axis lies in a given plane, is reflected obliquely at the centre of a concave spherical mirror; determine the spaces, within which if the point of divergence be situated, the reflected rays in the primary and secondary planes will both diverge or both converge; and shew that they are separated by a space, within which if the point be situated, the reflected rays in the primary plane will converge, and those in the secondary diverge. When the rays reflected in the primary plane are parallel, determine the locus of the geometrical focus of the rays reflected in the secondary plane. 153. If a pencil of parallel rays pass directly through a lens, the distance of the geometrical focus of emergent rays from the nearer focal centre in a convex lens, and from the farther focal centre in a concave lens, is the same on which ever surface the rays are incident. 154. If an eye be placed in air close to the surface of a clear stagnant fluid, prove that the apparent form of a circular arc in the fluid, whose centre coincides with the place of the eye, and whose plane is perpendicular to the surface, is defined by the equation m sin? 0 m2 - cos? O' where a is the radius of the circle, m the index of refraction, and the radius vector r is drawn from the place of the eye making the angle with the surface. 155. If the thickness of a concavo-convex lens be equal to (u + 1) times the distance between the centres of its spherical surfaces, shew that a point may be found in its axis, from which if rays diverge and fall upon the concave surface, they will diverge accurately from a point after emergence. a. = M 156. Determine the form of a small pencil of rays after passing obliquely through the centre of a lens. What are the advantages and disadvantages of limiting the aperture of a lens, and why is the object-glass of an astronomical telescope a lens of considerable aperture and small magnifying power ? 157. A ray of light passes through a double convex lens of small thickness; shew that if ε and n be the angles at which the incident and emergent rays are respectively inclined to the axis, c tann = 1+ и 1 + 2μ Au where the constant ratio of the T' sine of incidence to the sine of refraction. State also the nature and cause of secondary spectra. 159. The principal focus of a sphere bisects the distance between the focus after the first refraction, and the extremity of the diameter in the direction of which the rays are incident. 160. An object is placed between two plane mirrors inclined at an angle of 20°, find the locus of the images and their number. 161. A small pencil of rays is incident directly on a convex spherical refractor of a denser medium, find the focus of emergent rays. When the incident pencil is converging, determine for what positions of its focus the convergency is increased by the refraction. 162. Explain the construction and use of Hadley's sextant; when applied on land to determine the altitude of a star, how is the want of an accurate horizon supplied ? 163. Define the term critical angle; and prove that a pencil of rays cannot pass through a prism whose refracting angle is greater than twice the critical angle of the substance of which it is composed. 164. Find the magnitude and position of the least circle of aberration when a pencil of diverging rays is refracted directly at a plane surface. 1835 can 165. In the astronomical telescope, whether with a simple or compound eye-piece, determine the magnifying power; and breadth of the object glass shew that it is equal to the breadth of the emergent pencil How does this enable us to determine practically the magnifying power ? 166. Determine the distance at which a short-sighted person see distinctly, from observing the angle through which a glass prism is turned from its position of minimum deviation, in order that he may see distinctly through it a line of homogeneous light Where must an eye be situated, in order that when a luminous point is placed in the axis of a convex lens, the lens may appear wholly illuminated ? 167. To an eye placed at the aperture of the large mirror in Gregory's telescope there will appear an inverted image of both mirrors near the smaller; and if the axis of the smaller be slightly disturbed, the images will be shifted towards that part of it which is most inclined from the larger. Prove this property, and explain its use in the practical adjustment of the telescope. 168. If b, 6 be the breadths of the pth and the rainbows respectively, and 8 the sun's apparent diameter; shew that 6 = b+ (9 + 1)2 – M? . 8) nearly. (n + 1)^ - MỸ 169. If N, n, F, be the focal centres and principal focus of a lens, the distances of the conjugate foci measured from two points N', n', so situated in the axis that nn' = p.NN' р 1 (1 – p) nF, are connected by the equation 1 рп p being any constant quantity. 170. If a small pencil of rays pass directly through a medium bounded by concentric spherical surfaces whose radii are r, s; the equation between the distances of the conjugate foci measured from the centre is 1 1 V 171. Having given the focal lengths of two lenses, find that of the system formed by their combination when the lenses are placed (1) in contact, (2) at a given interval. Explain briefly the principle of the achromatic eye-piece. 172. A pencil of diverging rays is refracted obliquely at the centre of a thin lens ; find the distances from the centre of the foci of refracted rays in the primary and secondary planes. 173. Why is the form of the rainbow circular ? State the cause of the partial brightness of the sky within the bow. 174. Two lenses of equal focal length 3l are placed at a distance 21 from each other; required the forms of the lenses so as to throw the best possible image of a plane object on a distant plane surface, the pencils being defined by a diaphragm placed in the focus of the compound lens. Having given 1 {28x2 + 20 (a + B) x + 10aß + 3B2 + 27}. 175. Having given the focus of a pencil of rays incident directly on a spherical reflector, find the focus of reflected rays to the first and second approximations. 176. State the laws of reflection. Find the image of any straight line placed before a plane reflector. 177. Find the relation between the conjugate focal distances, when a small pencil of rays is refracted directly at a spherical surface. Apply the formula to find the focal length of a thin meniscus lens of glass, the radii of whose surfaces are 9 and 12 inches. 178. Explain the general principle of telescopes. What is the field of view of the astronomical telescope? Why are large instruments necessary for some observations ? 179. Determine the form of a surface on which, when rays diverging from a given point are incident, they may diverge after reflection from another given point. 1836 180. Explain the term “dispersive power” of a medium; and shew how it is measured. Find the conditions under which two prisms with small refracting angles will achromatize each other. 181. Rays issuing from a luminous point are incident upon a thin lens. A portion of those that enter the lens is allowed to proceed at once through the second surface ; a second portion, however, does not escape till it has been twice internally reflected; a third portion four times reflected; a fourth portion six times, and so on. Shew that a row of images will be formed at distances from the lens which are in harmonic progression. 182. A ray of light falls on the convex surface of a hemispherical lens in a direction parallel to its axis, is reflected at the plane surface, and emerges through the convex ; shew that the angles of incidence and emergence are equal, and that the distance between the points of incidence and emergence is equal to twice the deviation of the ray at either refraction. 183. A concave lens of glass is placed behind a given crossed lens of the same material, and joins closely with it; determine the radius of the outer surface that a pencil of parallel rays may pass directly through both lenses without aberration.* 184. Light passing through a lens of small aperture is received upon a screen placed at the geometrical focus ; determine the law of brightness of the image. What is the peculiar difficulty of solving this question, when the screen is placed in any other position ? for instance, what is the law of brightness of the least circle of aberration? 185. If y and s be respectively the radii of the first and second surfaces of a thin lens of given focal length, the longitudinal aberration will be the least possible when 1 1 2(u i 1) (1 1 1 1 + + and r r M + 2 S р M 186. The image of each point of a luminous object placed before a reflecting or refracting surface being a caustic, the whole image will consist of an indefinite number of caustics; The aberration for a pencil of diverging rays through a glass convex lens is 13 27 1 7.x2 + 10ar + 42+ where and 2f {722 u X |