348 



NATURE 



[August 9. 1900 



the generating circle), and describing circles of diameter 

 BE around various points on it. A line joining the point 

 of intersection of one of these circles with the epicycloid, 

 and the point of tangency with the mirror, will, when pro- 

 duced, give a reflected ray ; for example, JK produced 

 for circle described around H. The construction once 



cated by dotted lines), and measure around a corner each 

 time. 



About a hundred pictures are prepared for each series, 

 and the pictures then photographed separately on the 

 film, which, when run through the animatograph, give a 

 very vivid representation of the motion of the wave-front. 

 Three films have been prepared thus 



W^— ^ ^^-^ far — reflection of a wave entering a 



V /7^ J V ^^T^ I \.i .]] \{ \9 <^0"cave hemispherical mirror (Fig. 20) ; 



X^l^l^ \^ ' 2^ V L ' J ./ ^( ^ • jr reflection of a spherical wave starting 



"^**'''^ ^*ai— <^ in the principal focus of a concave 



hemispherical mirror (Fig. ui) ; and the 

 reflection of a similar wave within a 

 complete spherical mirror (Fig. 22). A 

 number of these constructions, taken at 



O v^ v:z^ O 



^^ K^ KJ KJ \J 



p^ 



r\ 



r~\ 



\^ \J \^ \J 



prepared, the series of wave-front pictures can be very 

 quickly made. Three or four sheets of paper are laid 

 under the construction, and holes are punched through 

 the pile by means of a pin, at equal distances along each 

 ray (measured from the orthogonal surface). 



The centre of the mirror and the point where its axis 





intervals along the film, are repro- 

 duced, and comparison of them with 

 the actual photographs shows the close 

 agreement between the calculated forms 

 and those actually obtained. 



I have already mentioned the fact 

 that the cusps on the wave-fronts trace 

 out the caustic surfaces. This is 

 beautifully shown in Figs. 23 and 24, 

 where the successive fronts are seen superposed. The 

 former is for the reflection of a plane wave in a spherical 

 mirror, the latter for the reflection of a spherical wave 

 starting at the focus of a similar mirror. The caustic 

 curve is shown by a dotted line in Fig. 23, and is seen to be 

 traced by the cusps on the wave-fronts. The construction 

 shows that there is a concentration of 

 energy at the cusp ; consequently 

 we may define the cusp as a moving 



f^ 



meets the surface are also indicated in the same manner. 

 The sheets are now separated, and corresponding pin- 

 holes are united on each sheet by a broad black line, 

 which represents the wave-front. After a time it becomes 

 necessary to consider double reflections, and to do this 

 we are compelled to construct twice-reflected rays (indi- 



NO. 1606, VOL. 62] 



focus, and the caustic as the sur- 

 face traced by it. Though I hesitate 

 in claiming that this relation, at once 

 so apparent, is at all novel, I may say that, so far as I 

 have been able to find, it is not brought out in any of 

 the text-books, caustic surfaces being invariably treated 

 by ray rather than by wave-front methods. 



The cinematograph series illustrating reflection inside 

 a complete sphere was the most difficult to prepare, as 



