DU. OLIVER LODGE ON ABERRATION PROBLEMS. 
787 
Now Oq and ^ are very nearly equal ; showing that diffraction really does depend 
on wave-length simply, in spite of motion of grating, so far as miniitise of the first 
order are concerned. 
But then this direction, (f), will not be the direction appreciated by the observer ; 
for the motion of his telescope will cause ordinary aberration, since his motion is 
partially across the diffracted rays. 
Not to confuse the figure, I indicate the telescope OP further along the ray. While 
the light is travelling along it its eyepiece will have time to move to Q, such that 
PQ _ V _ AA' _ 
OQ ~ V — AF ~ 
Hence, A'F is parallel to the axis of the telescope which receives the ray, and may 
be called the apparent or perceived ray. The angle at which it is inclined to the 
grating-normal may be called 6. 
Now 6 is less than Bq, and is very nearly the same as if wave-length had been 
1 ‘eally shortened to AD, instead of AC. 
Draw BE a tangent to the AD circle, and we get the wave-front appropriate to this 
shortened wave and a stationary grating; v/hile AE is the ray belonging thereto, the 
inclination of which to the grating-normal we may call [sin = (1 — a) sin d^.] 
Now, plainly, AE and A'F are very nearly parallel, but not quite; there is a 
second-order difference between 6 and which may be readily calculated. 
Perhaps the simplest way of displaying the result is to introduce the aberration 
angle POQ = e (such that sin e = a sin 0) and to write 
whereas 
sin 9 1 = HIM (f> — a cos (f) sin 9q ; 
sin 0 = sin (f) cos e — a cos (f) sin 6. 
(Or one might write cot 6 = cot (f> a cosec ^.) 
The difference between the apparent ray and the shortened wave-length ray is 
approximately 
{ 6 ^ — 0 ) cos 0 = sin (f) (1 — cos e) — cos (f) sin 0q, 
or 
0 — 0^ = tan 0 (cos ^ sin 0 sin (fy), 
and is probably quite too small to be detected. 
The point of the whole thing is that a grating has the same real effect whether 
moving or stationary, but that the motion of the observing telescope causes an 
aberration which necessitates very nearly the same alteration of its direction as if the 
waves were really shortened in simple proportion to the motion. The Doppler effect 
caused by motion of observer is, therefore, essentially a case of common aberration. 
5 H 2 
