316 



SCIENCE. 



[N. S. Vol. XVI. No. 399. 



NOTE ON THE MULTIPLE IMAGES FORMED BT 

 TWO PLANE INCLINED MIRRORS. 



Having noticed errors in the statement of 

 the number of images in a couple of text- 

 books, I have recently made a canvass of 

 forty-one books on optics or general physics, 

 and was surprised to find a general lack of in- 

 formation on this subject. Fifteen of the 

 books did not consider it at all, many of the 

 others took up only special cases where the an- 

 gle is 60° or 90°, and eight contained mis- 

 statements as to the number of- images. It 

 does not seem to be generally recognized that 

 the number of possible images depends upon 

 the position of the object, and that the num- 

 ber of these which are visible depends upon 

 the position of the observer's eye. As in only 

 five of the forty-one books examined was the 

 dependence of the number of images upon 

 the position of the object correctly stated, I 

 have thought it well to write out the following 

 analysis. 



Let <p be the angle between the mirrors, a 

 the angular distance of the object P^ from 

 one of them, P„ P^, etc., the images formed by 

 first reflection in the latter, P', P", etc., the 

 images formed by first reflection in the other 

 mirror. Then, if w be a whole number, the an- 

 gular distance from P^ to any image may ba 

 stated as follows : 



Pin. 2)1^ 



P"' 2(16 



p2» + i 2n^ + 2(<^ — a) 



If the p-th image fall behind both mirrors 



in set P.i,^ 



TT -\-<j) — a > 2n9 > IT — a 



in set Pj„ + i 



T + n > 2n(4 + 2a > T — + a 

 or 



^ + 0_a>(2« + l)0>7r — a 



These conditions are the same and may be 

 written 



or 



where the equality sign is brought in so that 

 p may represent the number of images in this 

 set as well when the last one falls on the line 

 of one of the mirrors. 



Similarly, if the r-th image fall behind both 

 mirrors in set P'" 



IT + a > 2»'/i > T — i/i -(- n 



insetP'^" + ' 



7r-|-9 — a>2»^ -f2(^ — «)>"■ — <^ 



n-\-ay{'2n+\)<p>-rr-<l> + a 



Hence 





and the total number of images ^^-=p-^-r. 

 If (f. divide evenly into w, p^r= izjcp and 

 the last images of the two sets coincide. Then 



2n- , 



=i'-t-' 



1=- 



and fc is independent of a. 



This much is correctly worked out in 

 Violle, Heath and several smaller works on 

 optics, but in no text on general physics in 

 English that I have seen. 



If the eye be behind the plane of one mirror, 

 only the images finally formed by the other 

 are visible. If the eye be between the mirrors 

 (angularly) all the images formed in front of 

 either mirror are visible. An image behind 

 both mirrors will be visible only to an eye 

 between the mirrors whose position makes an 

 angle less than tt — d with the surface in which 

 the last reflection takes place, where 6 is the 

 angle between this surface and the position of 

 the image considered. 



