228 
MR. AIRY ON THE THEORETICAL EXPLANATION OF 
y, and putting e 2 for c -f- a) 2 + b 2 , it is 
e _ A„_ + 
e y \2 ' Qc e) y ‘ 
With the small values of b, of which we shall have occasion to treat, b 2 will be much 
d (& 
less than — . Neglecting therefore the first multiplier of y 2 , and putting § for the 
distance from the point x, y to the point on the retina, 
b a 9 
t = e--y- iT e y 2 
cb 2 a ( cb \ 2 
“ e + 2^~2 Tew 
where/ = e + 
If now we suppose (as in diffractions generally) that every part of the great wave, 
after leaving the lens, is simultaneously the origin of a small wave diverging in all 
directions, at least through a large angular extent, we may represent the disturbance 
of ether on the point of the retina, produced by the small portion ty of the large 
wave, by 
q _ 
S) 
ly X sin (v t 
or 
or 
Sy X sin x {»<—/+ 2^7 (y + ~) 2 } 
• 2,r / . /-\ v 2 7T 
sin — (y t — j ) . I y cos — 
h + ^ 
£ c e 
i 2 Ti- „ v. • 
+ cos -r-(vt—j).oy sin 
and consequently the disturbance of ether on the point of the retina produced by the 
whole wave will be 
sin (v t -/) / cos ^ (y + ^) 2 + cos / (v t -/) f y sin 3^7 (y + ^)'- 
The limits of y in the integral must be the limiting values which determine the extent 
of the great wave where it leaves the lens under the circumstances assumed, that is, 
as far as there is no additional cause of retardation of the wave. 
But if, by the interposition of any refracting substance with parallel boundaries, a 
portion of the wave be retarded by the phase R (expressed as an angle), then the ex- 
pression to be integrated will be 
/sin (x {”<-/+ 2^7 h + VT} - R ) 
