MR. R. A. SAMPSON; A NEW TREATMENT OF OPTICAL ABERRATIONS. 153 
In the diagram Pf, is the point where the original and emergent rays meet at the 
surface, PPo is the original ray, P'P,, the emergent ray, and (O, h, c) are the co¬ 
ordinates of P, (O, 6', c') those of P', and we shall take {a^, Cq) as those of Pq. 
If {I, m, n) {I', m', n') are the direction cosines of an original and emergent ray, 
{p, q, r) those of the normal to the surface at the point of incidence, we have the 
known equations 
{fil — fx'l')!]) = {ium — /jfm')lq = {/u(.n — /jL'n')fr = /j. cos d — ij! cos 0', 
where 0, 0' are the angles made by the two rays and the normal. 
Now 
I = w//3 = n/y = 1 -W-^y 
= =n'ly' 
-p = (7/B6o + iC5o(&o" + Co") = '>7 Bco + ^Cco(6,/ + c/) = -1 -f-i-BV + iBV = - l+W + h'"^ 
if we neglect higher powers of the small quantities. 
Further 
cos 0=1 —^0^, 
where 
ff‘ = {/S-qy + {y-ry, 
and we have approximately 
m(/3-Q') = ^{(3'-q), 
cos e' = 
= {l3'-qY + {y>-rf, 
fi{y-7') = ix'{y-r). 
Substituting above for m, 7n\ we have 
But since 
O/fi' = d'/fi, 
VOL. ccxii.—A. 
X 
