REFLEXION AT A PLANE SURFACE. When a virtual imag,e of luminous object is formed by reflexion at a plane surface, there is no spherical aberration in the pencils, nor any distortion in the image, and the imag,e is situat,ed in exactly the same relative position with respect to the reflector, behind it, as the luminous object in front of it. As these points should be clearly understood we shall demonstrate them with the help of a diagram.
Let P Q be a luminous object placed before a plane reflector ED ; and let l'C be any one of the rays of the pencil proceeding from P. This ray after reflexion at C will follow a course C R such that CR and PC make equal angles with CD. Draw the line PA perpendicular to the plane of the re flector, and produc,e it top, making Ap =AP. Join p C. Then in the tri angles PAC, pAC, which lie in the same plane, pA=PA, CA is common to both, and the included angles at A are right angles, therefore the angle pCA=PCI. But theaugle ECR=
PC A, and is in the same plane with it ; therefore p C A=ECR ; CR is consequently in the same straight line with Cp. Hence it follows that the reflected ray CR, if produced backwards, passes through p. But the position of the point p does not depend upon the distance AC, or the angle P CA ; it is therefore the same for every reflected ray of the pencil from P. Therefore p is the virtual image of P, and the reflected pencil is entirely free from aberration.
In the same way it may be shewn that if Q be any other point of the object, and Qq be drawn perpendicular to the reflector, B Q being equal to B q, q is the virtual image of Q.
Hence it follows that pq the virtual image, and PQ the object, are symmetrically situated with respect to the plane of the reflector DE, on opposite sides of it.