798 
DR. OLIVER LODGE ON ABERRATION PROBLEMS. 
Lastly 
BC sin (t -f e) . AE sin {%' — e) 
AC cos e 
AC 
cos e 
These solve the problem, and they may be conveniently worked on the following 
lines— 
sin e = a sin {^ ~ (/>) ; sin e = a sin [i + (f)), 
sin i — cos i tan 
e' _ AE _ V cos e' + V cos 8' _ cos e' — a cos (i' + (j)) _ cos e — a cos (i — 
sin i + cos i tan e EC V cos e + v cos 6 cos e + a cos {i — <p) cos e' + « cos {i' + ^)' 
the last equality being added for convenience, and being true because 
cos^e — cos^ 6 ~ 1 — 
Therefore, exactly, 
sin i cos e — ^ a” cos i sin 2 [i — <^) sec e = sin i' cos e — cos i sin ^ {i' + </>) sec e, 
whence, expanding cos e and neglecting 
sin i — sin i' = cos^ i sin 2(f), 
or 
i — i' := ot^ cos^ i sin 2(f). 
The discrepancy between the angles of incidence and reflexion (which I call for 
brevity the error of reflexion) is therefore exactly expressible in even powers of 
aberration magnitude, and no part of it reverses with the reversal of the ray. It 
vanishes for grazing incidence, and is a maximum for normal incidence (at which I 
am somewhat surprised). It vanishes both for tangential and for normal drift, being 
a maximum when the medium drifts at 45° to the mirror. 
The maximum possible value of the error of reflexion is a'^, or 10 “® of a radian, or 
0"‘00205, or y^ol^h of a second of arc ; an amount which, although equivalent to an 
error of only 15 inches in the circumference of the earth, it is perhaps possible to 
detect; especially if, as Mr. Michelson suggests, it be increased by multiple 
reflexion. Indeed, it strikes me as perhaps the simjjlest way of examining into the 
motion of the ether near the earth. 
Refraction in a Drifting Medium. 
69. The reasoning for refraction is precisely of the same kind, and there needs 
nothing more but to wu’ite down the equations, putting V/p. everywhere instead of V, 
vjgd instead of v, and consequently a p instead of a. 
It is best thus to assume Fresnel’s theory, and leave observation to point out any 
deviation from it that may be existent. 
