OPTICS. 822. If p be the number of reflections, A the of incidence, and B that of refraction; then the between the incident and emergent ray is (Wood, Prop. XCIII.) 202 (p+1) '. Also for the rays to emerge parallel, we must have (Wood, Prop. XCIV.) and putting its differential = 0, we get 4 cos. (cos. ' — ' sin. 0') = cos. ' sin. Hence, by substitution, we get, after the proper reductions, 823. Let r be the radius of the hemisphere, x and y the co-ordinates, measured from the edge of the bowl, of the nearest point which comes into view when it is filled with water; then it readily appears that the depth (d) of the image of this point; dy: sin. ' ; sin. 0 :: 3 : 4 and 'being the angles of incidence and refraction. 824. If A and A' denote the distances of the foci of in cident and refracted rays, we have (Coddington, p. 56) and this substituted in the expression for the maximum gives 825. Since the object is small it may be considered a circular arc concentric with the sphere, and its image after reflection will therefore be a circular arc, also concentric with the sphere, and similar to the object, (Wood, Prop. XV.) Again, if ▲ denote the distance of the image from the refractor, in order that its image, caused by refraction, may be distinct, we must have (Wood, Prop. XXXIX.) Arr: sin. I sin. R :: 4 : 3 Hence the distance of the image by reflection from the reflecting surface is and it is.. exterior to the sphere; and the distance of the object from the same surface is. (Wood, p. 26,) Hence the distances of the object O and image O' by reflection Again, since the distance of the image by reflection from the reflecting surface is the distance of its image caused by refraction from that surface is got from the expression Hence the distance of this image, which is concentric and similar to the former, from the centre is Now let a be the distance of the eye from the centre; then its Also, since the distance of the principal focus of the sphere from the centre is (Coddington, p. 66,) .. the visual angle of the object when at the principal focus is .. 0 : 0′ :: a + 2r: 250 × 7. (26r — a). 826. upon From the centre of the reflector let fall a the rectilinear object produced; then taking this as an axis, the image will be a portion of a conic section, whose major-axis coincides with that L, and whose focus is the centre of the reflector. Let be the traced angle, and p the radius vector of this conic section, c the distance of the foot of the object from the centre, and the radius of the reflector; then (Coddington p. 36,) and the equation between its rectangular co-ordinates is known, viz' Again, let & be the inclination of the object to the given axis, d its length, and n the part of it produced. Also let denote the inclination of the tangent at any point of the image to its major axis; then |