flare spots in photography. 215 
Now, by the methods of § 6, we have 
SKkK=u+de/u, TL =v, + th?/v, 
SX =SM—-AM=u-4?/a, T,Y,=T,N— DN =», — eld. 
Hence 
AK =SK — SX = th? (1/u + 1/a) 
LY, =T,L — T,Y, = 4h? (1/v, + 1/d). 
We have also 
AB=th? (1/a+1/b), KL = $h? (1/b + 1/c), CD= 1h? (1/c + 1/d). 
When we equate the optical length of the path wia K to that 
via A in each of the six cases, we obtain the following six 
equations; the symbols [DC], [DB]... indicate the two faces which 
have acted as reflectors*. 
[DC] TUR fe ROY JN, = PAB +3nCD 
[DB] RRL ID ey Os AUB SMO 
[DA] NIC Th TUNG, NG yn US OD) 
[CB] NG OeD enV OED =, ABE i OD) 
[CA] TOC A Fh 4 J 4 DIRT SDS WOE 
[BA] STR a GD 2756 =3yAB+ wCD 
Let the primary focal lengths of AK B, CLD be m,n cm. and let 
the corresponding powers be M, N cm... Then 
(w—1)(1/a+1/b)=1/m=M, (w—1)(1/e+1/d)=1/n=N. 
Let the secondary focal lengths of AA B, CLD be m,, n, cm. and 
let the corresponding powers be M,, N, cm... Then, by § 7, 
(84-1) Ga +1/b)=1/m,=M,, Bp’ —1) Ale + 1/d) = 1m, =N,. 
Let — 2(1/b + 1/c) =1/w = W. 
It is worth noting that w is the secondary focal length of a “thin” 
lens of refractive index unity and radii 6 and c, the surfaces being 
concave in the case of Fig. 5. The primary focal length of such 
a lens is infinite. An optical system formed by air enclosed 
between two spherical soap films is a very close approximation 
to such a lens. 
Let f, be one of the six secondary focal lengths and F, the 
corresponding secondary power of the system, so, that 
Ji, = Wij = Wien Fel. 
Then, when the six optical equations are multiplied by 2/h?, we 
obtain the following values for the six secondary powers:— 
* A more general method, applicable to any number of thin lenses in contact, 
is given in § 14. . 
