SPHERICAL ABERRATION FOR A SYMMETRICAL OPTICAL SYSTEM. 
45 
The value of C, when written out fully, is given by 
0 = f (n 2 —n 0 )~ 2 { — {n 2 —n 0 n a + n 0 2 )-\-(n 0 2 + n a ) 'Ml l + (n 0 2 —3n {) n 2 + n a 2 ) M, 2 }. . (32) 
§ 6. The Convergency Factor and the Singular Inclination for a Single 
Refracting Surface. 
Having now obtained expressions (24 and 26) for the longitudinal spherical 
aberration, which are correct to the second order of aberrations, when expansion 
in powers of sin a 0 or tan a 0 is legitimate and rapidly convergent, we have now to 
enquire how far the same expression remains valid as M increases, in which case we 
know that B or B increases without limit and the convergency fails, even for 
comparatively small values of a 0 . 
Here it will be convenient to introduce two definitions :— 
I. We shall call singular inclination the value A (see § 3) of a 0 for which the 
emergent ray is parallel to the axis. 
II. The factor 1 — sin 2 a 0 /sin 2 A (or 1 — tan 2 a„/tan 2 A if we are dealing with tangents) 
we shall call the convergency factor. If we multiply Ax by the convergency factor 
we remove those singularities of Ax which are instrumental in causing critical failure 
of convergency. 
To find the singular inclination and convergency factor for a single refracting 
surface, we have to find when a 2 = 0. 
Going back to the fundamental equations (8) to (ll) we have a 2 = 0 when a 0 = A, 
where 
A- f a l/'-y 
which leads to 
sin A = sin f 2 cos —sin \fr 0 cos f 2 
= (n 0 x o sin \/n 2 rJ^/ {1 — (x 0 sin X/rf} - (x Q sin A /n) a/ {1 - (n 0 x 0 sin A /n^) 2 }. 
Hence, either sin A = 0, which obviously refers to the axial ray, a trivial and (for our 
purpose) irrelevant solution, or 
rfx 0 = (n„/ri 2 ) v /{l-(x 0 sin A/7- 1 ) 2 }- x /{l-(n 0 a- 0 sin A/n 2 r,) 2 }.. . . (33) 
On rationalising (33) leads to 
4 n 0 2 sin 2 A /n 2 = 4 r x 2 /x 0 2 - ( 1 + r^/xj - n 0 2 /n 2 2 ) 2 
= -(1 +r i /x 0 -njn 2 ) (l +r 1 /x 0 + njn 2 ) (l -rfv 0 + njn a ) (l -rjx 0 -njn 2 ). (34) 
This gives the singular inclination. 
If we write 
B — (1 T / j /Xy H(|//i 2 ) (1 + >'Jxu + njn 2 ) (l rtjn 2 ) (I > \In ^ 0 /^ 2 ), • (45) 
VOL. CCXXI.—A, 
H 
