I 
DR. OLIVER LODGE ON ABERRATION PROBLEMS. 
797 
67. It is easy to see that the triangle PA'C is isosceles, and accordingly that the 
angle A'GA is equal to half the difference of the inclinations of incident and reflected 
waves to the mirror surface ; i.e., calling this angle 77 , 
277 = {i + e) — {i — e), 
or 
e + e i — 
^ 2 ^ 2 ’ 
hence the wave is reflected precisely as if the mirror were rotated through the angle 
77 and there were no drift ; the angle of virtual rotation being very approximately 
the mean of the abeiTation angles. 
Fig. 18. 
The first approximation to its value is 
77 = a sin i cos </>; 
it practically vanishes, therefore, for normal incidence and for tangential drift. 
Further, as regards the change of width of the beam or distance between the rays, 
it is apparent that measured along the wave-surface it is the same, because 
EC = A'B' = AB ; so measured perpendicularly it changes in the ratio of cos e : cos e' 
before and after reflexion. 
68 . It is not to be supposed that the ray is reflected after this manner ; and, in fact, 
we shall find that the error of ray-reflexion, or difference between angle of incidence 
and reflexion, i — F, is exceedingly small. 
To determine this difference, and the whole circumstances of the problem, we write 
down the following equations, obvious from figure 17 : 
sin e v A A' sin e 
Sind ~ B^ ~ V ~ ^ ~ A'E “ siu0' ’ 
6=1 — (f), 6 ' — TT — <A)- 
Also, for the time of journey of the wave from position AB to EC, 
BB' B^_ BC _ AE _ A'E AA' 
V V cos € + V cos 6 V cos e + v cos 6' V 
V 
V 
