106. A reflector is formed of an indefinite number of faces, which are tangents to a parabola, and rays diverge from a given point in the axis of the parabola; find the locus of the images produced by reflection. 107. A small pencil of rays being refracted obliquely at a convex spherical surface, determine the foci in the primary and secondary planes. 108. Determine the aberration of a pencil of parallel rays refracted directly through a double convex lens; and if the index of refraction be , find the ratio of the radii when the aberration is a minimum. 109. Determine the relation between the focal length of two lenses which shall achromatize each other when separated by a given interval. 110. A ray of light, which passes through a prism in a plane perpendicular to the axis, is turned towards the thicker part of the prism, if it be denser than the surrounding medium. 111. If a luminous point be placed between two plane mirrors inclined to each other, prove that there will be a series of images which lie in the circumference of a circle: determine also the number of images. 112. How does it appear that light emanates in straight lines? and how has its velocity been determined ? 113. When a small pencil of diverging rays is incident on a concave spherical reflector, prove that FQ: FA :: FA : Fq. Shew that the foci Q and q are on the same side of F, and move in opposite directions. 114. Describe the astronomical telescope in its simplest form, and determine its magnifying power. What are the obstacles to the perfection of refracting telescopes ? and what means have been taken to obviate them? 115. If a small cylindrical pencil of rays fall obliquely on the centre of a spherical mirror, the lengths of the focal lines in the primary and secondary planes are to one another as 1 : cos'o, where ø is the inclination of the axis of the pencil 1832 to the axis of the inirror. Also the diameter of the circle of least confusion is 1 (1 - cos°°) where is the diameter of 1 + coso the circular section of the incident pencil. 116. In the astronomical telescope state fully the defects of the simple eye-glass. How are they remedied by the achromatic eye-piece? 117. A small pencil of rays not parallel is incident obliquely on a plane refracting surface; determine the foci in the primary and secondary planes. 118. Explain the formation of the image of a portion of a straight line placed before a convex spherical reflector, compare the linear magnitudes of the object and image, and draw the course of the extreme rays from the object to an eye in a given position. 119. If a straight rod be immersed in water in a horizontal position, shew that it will appear to be lengthened; and if it be immersed in a vertical position, that it will appear to be shortened. 120. A slender rod partly immersed in water contained in a vessel, casts a shadow on the bottom of the vessel from a light placed near it; it is observed that the shadow of the part of the rod immersed is separated from that of the other part by a bright interval; shew that this is a consequence of capillary attraction. 121. At a given hour on a given day find the inclination of a plane reflector to the horizon and meridian, that it may reflect the sun's rays in a given horizontal direction. 122. A small pencil of parallel rays falls directly on a double concave glass lens of equal radii, and after refraction at the first surface is reflected at the second; shew that it will diverge from a point at a distance from the surface equal to of the radius, neglecting the thickness of the lens. 123. A pencil of diverging rays is incident directly on a double convex lens; find the focus of refracted rays to the second approximation. 124. Determine the exterior colour of the primary rainbow, and shew that the order of colours in the secondary bow is the reverse of that in the primary. 125. The deviation of a ray passing through a prism is a minimum when the angles of incidence and emergence are equal. Shew how this proposition is used to find the refractive power of any substance. 126. A mixed pencil of parallel rays passes eccentrically through two lenses of given focal lengths ; determine the interval between the lenses that the directions of all the partial emergent pencils may be parallel. more! 127. A pencil of parallel rays passes directly through a lens; determine the diameter of the least circle of chromatic dispersion. 128. Explain the method by which Newton proved the sun's light to consist of rays which differ in refrangibility and colour; and that each ray of the solar spectrum, when once separated by refraction, does not admit of farther separation. 129. Find the focal length of a lens of inconsiderable thickness; and shew that a short-sighted person requires concave glasses, and a long-sighted person convex ones. 130. A small pencil of rays is incident nearly perpendicularly on a convex spherical mirror, determine the geometrical focus of reflected rays; and shew that the increase of divergency, or decrease of convergency, is the same for all pencils. 131. Describe the eye; why do objects viewed at near distances appear enlarged and distorted ? 132. Parallel rays refracted at a plane surface continue parallel. 133. Define a ray of light, and state the experiment which proves that if the refracted ray becomes the incident, the inci. dent becomes the refracted ray. 134. Describe the astronomical telescope in its simplest form, tracing the course of the extreme rays. How is a uniformity of brightness secured in the field of view ? What are the additions requisite to make the telescope a good one? 135. When a small pencil of diverging rays is incident obliquely on a plane refracting surface, determine the foci after refraction in the primary and secondary planes; and shew that 1833 after refraction through a prism, these foci will coincide, if the angles of incidence and emergence are equal. 136. A given small luminous object is placed at a given point in the line passing through the centre and perpendicular to the plane of a circular area. Compare the illumination of the whole area with that which it would receive if the light thrown on it was uniformly the same as that at the centre. 137. Parallel rays are incident upon a cylinder in a direction perpendicular to its axis ; shew that the equation to a section of the caustic surface is (? — 1) y = {(u?a); – x}}} + {aš – (2x)}}}, the centre of the corresponding section of the cylinder, whose radius is a, being the origin, and its diameter, perpendicular to the incident rays, the axis of x, and u the index of refraction. 138. A small plane reflector stands upon a horizontal plane and inclined at a given angle to it. Determine how great a length of his own person a man standing before it at a given distance from it can see, and where his position must be that he may see the greatest possible length. 139. A small pencil of diverging rays is incident directly on a concave spherical refractor; to investigate a formula for determining the focus of refracted rays, giving the first and second approximations. 140. When a ray of solar light is refracted by passing through a prism, describe the appearance of the spectrum when seen in its state of greatest purity. How do different media affect the positions of the fixed lines in the spectrum ? Are the phenomena the same for all kinds of light? 141. A small pencil of diverging rays passes directly through a lens, required the diameter of the least circle of chromatic dispersion. 142. The general formula for direct refraction through a double convex lens being 1 1 u + (u - 1) + + 2u? + + + u u V u S rs 1834 to determine the ratio of r:s, when the aberration is a minimum. 143. The distances of the conjugate foci from the first and second focal centres of a lens are connected by the equation 1 1 It (u 1) M 144. Rays tending to form an image in the axis of a concave lens are refracted to an eye situate in that axis, find the visual angle; and determine the position of the image when the visual angle is the same for all distances of the eye. 145. Explain the experiments by which it appears that when a ray of light passes out of one medium into another, the sine of incidence is to the sine of refraction in a given ratio ; and that if the refracted ray becomes the incident, the incident becomes the refracted ray. 146. Investigate, geometrically or algebraically, the relation between the distances, from the surface of a spherical reflector, of the conjugate foci of a pencil incident nearly perpendicularly upon it. Adapt the formula to diverging or converging rays, drawing the requisite figures. 147. Describe the construction and use of Hadley's quadrant, and state why the reflectors are at right angles to its plane. What is meant by the index error ? 148. Explain the construction of Newton's telescope, tracing the course of a ray. What are the defects attending the use of a simple eye-glass in a telescope ? 149. A given small pencil of diverging rays is reflected obliquely at the centre of a spherical mirror; shew how to determine the form of the reflected pencil. Explain the term focal lines ; and, having given the foci of reflected rays in the primary and secondary planes, determine the position and magnitude of the circle of least confusion. 150. A small pencil of parallel rays passes directly into a sphere of water and is reflected at the second surface, determine the geometrical focus of emergent rays. 151. The ends of a glass cylinder are worked into portions of a convex and concave spherical surface, radii r, s, respec |