44 



UNDULATORY FORCES. LIGHT. [LAWS OF CURVED REFLECTOM. 



\ 



Referring to Fig. 5, if a 6 represents a spherical con- 

 cave reflector, c is the centre of concavity, or that point 

 from which, if straight lines are drawn, they shall touch 

 the mirror, and be of equal length ; they are, in fact, the 

 radii of the circle, of which the mirror is but a portion : 

 e d is the axis of the mirror : e d represents a ray of 

 light passing along the axis, and incident on the mirror 

 at d. This ray, because passing along the line of the 

 centre of the mirror to d, will be reflected back in the 

 same straight lino. But if another ray, g h, impinges 

 on the mirror at h, it will not be reflected back in the 

 same line. The direction which such a ray will be 

 reflected in, will be towards /, and will be in the line h f. 

 This will be evident if a line be drawn from h to c ; for, 

 as the angle g h e, which is the angle of incidence, is 

 equal to the angle c h f, which is the angle of reflection, 

 the law of reflection is at once satisfied ; and the point 

 f, in the line of the centre of concavity, becomes the 

 focus of the mirror, and is situate half-way between the 

 centre of concavity and the centre of the reflector at 

 d. This may be better understood to the general reader, 

 if we deal with the subject in the following form. (See 

 Fig. 6, in which we employ the same letters as in the 

 last engraving). 



Supposing g h to be a ray of light touching an ex- 

 ceeding small plane, forming part of the sphere, and 

 Fig. 6. that the line h c is perpen- 



J,/ dicular to h. Now the angle 



~y of incidence, g h c, must be 

 equalled by the angle of re- 

 flection. To find the latter, 

 draw the line h f, so that 

 /shall be any point on the 

 straight line c d ; and in so 

 doing, let the angle c h f 

 equal the angle g h e. It will be found, that the point 

 / is necessarily half-way between c and d, and, as we be- 

 fore stated, is the focal point of the mirror. This illus- 

 tration can easily be repeated, by the non-mathematical 

 student, by a pencil on paper ; and that plan will assist 

 materially in relieving the difficulty of understanding 

 this portion of the laws of reflection. 



It also follows, that all parallel rays impinging on the 

 mirror, except at the point d, will necessarily be nearly 

 focalised at/; because, with every ray, the circumstances 

 are almost exactly the same. To understand this, sup- 

 pose the plane h to be moved nearer to d, and that a 

 ray of light impinges on it in a straight line. In such 

 case, a fine perpendicular to its new position must be 

 drawn from it to c, the centre of concavity ; and as the 

 same course of reasoning must be adopted to explain the 

 results so obtained, the focus of the parallel rays will 

 still be found at /. 



As an application of the above remarks, we may state, 

 that an image of a star, if its light were thus reflected 

 by a spherical mirror, would be found at /. 



We may, at this place, urge on those of our readers 

 who may not be accustomed to strict scientific reasoning, 

 that they should not pass any fact or illustration without 

 having thoroughly mastered its details. In every branch 

 of science in which mathematics are employed, it is neces- 

 sary that succeeding facts should have a relation to those 

 already explained. A chain of argument is thus formed ; 

 and if one link be broken, the whole arrangement becomes 

 defective. ' ' Slow and sure " should be the guiding maxim 

 of every tyro in philosophy. 



If the source of light be placed in the focus of such a 

 mirror, all its rays would be collected on the surface, 

 and be reflected therefrom in straight lines. This will be 

 at once evident, on considering that by such an arrange- 

 ment we simply invert the progress of the rays by send- 

 ing them from, instead of to, the focus of the mirror. 

 If the source of light were nearer to d, the centre of the 

 mirror, than to /, its focal point, then the rays would 

 not proceed in parallel lines from the reflecting surface, 

 but would diverge outwards, because the angle of inci- 

 dence would bo greater than those which are produced 

 when the luminous object is placed in the focus ; and as 

 the angle of reflection must equal it, that must be 



greater also. This will be evident on inspecting the 

 following diagram. 



Using the same letters as before, and presuming x 

 to represent the new position of the source of light be- 

 tween / and d, wo ob- Fig. 7. 

 serve, that if a perpendicu- 

 lar is drawn from h to c, 

 the angle x h e is greater 

 than the angle e h f. Now A 

 as x h c, the angle of inci- 

 dence, must equal the angle 

 of reflection c h i, it fol- 

 lows that the ray of re- 

 flected light will diverge as 

 at h i; and thus the fact 

 of the divergence of the rays from x, is at once made 

 evident by comparing the direction of h i with g h, 

 the parallel ray. 



The foregoing illustration of the properties of rays 

 reflected from a spherical surface, may possibly be dif- 

 ficult to a beginner. We have been compelled to use a 

 circumlocutory description, because we presume that some 

 of our readers will be unacquainted with geometry. To 

 a mathematician, the proposition will appear self-evident ; 

 and we trust that, after a little study, others may find 

 the matter easily understood. In every case of reflec- 

 tion, the student should bear in mind the laws of inci- 

 dent and reflecting angles, and, by strictly applying 

 them, many of the problems of optical science will be 

 readily apprehended. 



We now proceed to investigate the results which are 

 arrived at when rays of light fall on a spherical mirror 

 from objects at but moderate distances from it, and in 

 lines which are not parallel to each other. To simplify 

 our observations, we shall suppose that a ray of light, 

 from any source, falls on the reflecting surface, as repre- 

 sented in our annexed diagram. (See Fig. 8). 

 Fig. 8. 



In our illustration (Fig. 5), we presumed that the source 

 of light was at an infinite distance from the mirror, and 

 that all these rays were consequently reflected so as to 

 meet in one point distant from the reflector, and called its 

 focus. In Fig. 8, we suppose the case of the source of 

 light being placed at s, and s ax to be the axis of tlio 

 mirror ; ce the centre of curvature. Let p bo any piano 

 in the mirror on which a ray s p shall fall. The lino 

 p ce being perpendicular thereto, the angle p ce f will 

 be the angle of reflection, and the focal point of this 

 ray will be found at /. But suppose another ray of 

 light, s p', to fall on the mirror at p', then 3 p' ce will be 

 the angle of incidence. Draw a line, p' /', so as to form 

 an angle equal to s p' ce; then, as the angle ce p' f 

 3 p' ce, which is the angle of incidence, ce p' f will bo 

 the angle of reflection. But /' is not at the same point, 

 in the line s ax, as/, but is nearer to ce. Hence it is evi- 

 dent that all the rays passing from a source of light at 

 will not converge to one point, and of course, therefore, 

 their focus will be removed from that place which would 

 be found between ce and ox if all the incident rays 

 were parallel. This produces what is termed aberration, 

 or what wo may call, in plain language, the straying of 

 some of the reflected rays of light from one converging 

 point, and, consequently, the production of several foci 

 instead of one. The optical effect which tliis produces, 

 we shall more fully inquire into presently. 



As the reflectors to which we have referred have been 

 assumed to bo portions of a sphere, we generally term 

 the effect of the departure of rays from a common focus 

 as spherical aberration. It would take up too much of 

 our space, and we should have to presume a greater 

 amount of mathematical knowledge on the part of some 



