flare spots in photography. 219 
If W,,=—2(1/b, + 1/a.), Wo3=— 2(1/b, + 1/as), ..., we have 
GIy = OP Vio, IEG a at UI oe nO 
Further, as in § 11, 
XK, = tA /u+1/a), YK, =4(1/v + 1/b,). 
Now let = denote summation for all the quantities of any type 
and let S denote summation for only those quantities which refer 
to spaces through which a ray has passed three times. 
The optical length of the path from X to Y is equal to that 
from A, to B, (see § 11) and hence, if we omit the common factor 
+h, the general optical equation becomes 
Wee Sines tg 
Oh: Git Pee aN Oo w—l w—1 
2 
or +. +5(-+;)-8W=> oe fue ae 
Ww v @ lb wool. ~ p-l 
Since for any lens 
alah Coe 
Gad 
we have 
5 ME -=(- == 
= ad mee 
and hence 
ee Ss se Si Bae mates as (17). 
Ue) j= Il 
For any lens 
2uF Byu-1 
P+ = F'=secondary power of lens. 
w-l p-l 
Further, by § 11, Wi, W.s;, ... are the secondary powers of the 
successive air spaces, and hence the result (17) can be expressed 
in words as follows: 
Secondary power of system={Sum of primary powers of 
lenses traversed once} + {Sum of secondary powers of lenses 
traversed three times} + {Sum of secondary powers of air spaces 
traversed three times. 
The six secondary powers found in § 11 for a system of two lenses 
are particular cases of this general result. 
If the lenses are so chosen that the primary power of each is 
positive and the value of every W is positive, all the secondary 
powers of the system will be positive. 
When the lens used in § 10 is placed between the lenses used 
in § 13, the fifteen secondary images are easily seen. 
