344 



NATURE 



[August 9, 1900 



accustomed to say that if we confine ourselves to a small 

 area around the axis, a mirror of spherical form acts 



almost as- well as a parabola. If on the contrary we 

 consider the reflection from the entire hemisphere, we see 

 that the reflected wave curls up at the edges, 

 having^ a form not unlike a flat -bottomed saucer. 

 The flat bottom moves straight up, travelling 

 everywhere normal to its surface ; but the curled 

 up edges converge inwards, coming to a focus in 

 the form of a ring around the flat bottom. This 

 ring, of course, does not show in the photograph, 

 which is a sectional view, but it will be seen that 

 in one of the views (No. 4) the curved edge has 

 disappeared entirely. In reality it is passing 

 through a ring focus, and presently it will appear 

 again on the other side of the focus, curved the 

 other way, of course, and trailing along after the 

 flat bottom. This curious evolution of the wave 

 can be shown by geometrical construction, and I 

 shall show later how its development can be 

 shown with the cinematograph. 



When the spherical waves start in one focus 

 of an elliptical mirror, they are transformed by 

 reflection into converging spheres, which shrink 

 to a point at the other focus, the surface being 

 aplanatic for rays issuing from a point. An ellip- 

 tical mirror was made by bending a strip (Fig. 6) 

 of metal into the required form, and a soundwave 

 started at one of the foci. The transformation of 

 the diverging intoaconvergingsphere,and the shrinkage of 

 the latter to a point at the other focus, is well shown (Fig. 6). 



mirror, those reflected from points of the mirror near 

 its axis converge approximately to a point situated half- 

 way between the surface of the mirror and its centre of 

 curvature. The wave-front in the case of parallel rays 

 is, of course, plane, and is changed by reflection into a 

 converging shell of approximately spherical curvature. 

 If we investigate the case more carefully, we find, how- 

 ever, that the reflected rays do not come accurately to 

 a focus, but envelope a surface known as the caustic — 

 in this case an epicycloid. The connection between the 

 wave-front and the caustic is perhaps not at once 

 apparent. Let us examine the changes wrought on a 

 sound-wave entering a concave hemispherical mirror 

 (Fig- 7)- 



If we follow the wave during its entrance into the 

 mirror, we see that the reflected portion trails along 

 behind, being united to the unreflected part at the 

 mirror's surface. After the reflection is complete, we 

 find the reflected wave of a form not unlike a volcanic 

 cone with a large bowl-shaped crater (No. 4). This 

 bowl-shaped portion we may regard as a converging 

 shell, which shrinks to point at the focus of the mirror. 

 As it shrinks, the steep sides of the cone run in under 

 the bowl, crossing at about the moment when the con- 

 verging portion is passing through the focus (No. 6). 

 The rim of the crater forms a cusp on the wave-front, 

 and if we follow this cusp we shall see that it traces the 



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Fig. 6. . 1 



We will consider next another case of spherical aber- \ 



ration. When parallel rays of light enter a concave ' 



NO. t6o6. vol. 62] 



caustic surface. Hence we may define the caustic as 

 the surface traced by the cusp of the wave-front. 



The portion of the wave which comes to a focus at 

 once begins to diverge again, uniting with the sides of 

 the crater, the whole moving out of the mirror in a form 

 somewhat resembling a mushroom or the bell of a 

 Medusa jelly-fish. The turnedunder edges of the bell 

 are cusped, and these cusps trace the caustic enveloped 

 by the twice-reflected rays. These forms can also be 

 constructed geometrically. 



A much more complicated case is now shown (Fig. 8). 

 Here the wave starts within a complete sphere, or rather 

 cylinder. (Cylindrical surfaces have been used in all 

 these cases for obvious reasons, the sectional views 

 shown in the photographs being the same for both forms 

 of surface.) Starting in the principal focus of the closed 

 mirror, the wave is bounced back and forth, becoming 

 more complicated after each reflection, yet always sym- 

 metrical about the axis. Only a few of the many forms 

 are shown, and, with the exception of the first three or 

 four, are not arranged in order ; for at the time that the 

 series was arranged on the slide this case had not been 



