10 PROCEEDINGS OF THE CANADIAN INSTITUTE. 
8. The principal foci F, F’ of a system of two surfaces S,, 8, con- 
stituting a lens may be found as follows :-— 
Let (F,, F’,), (F., F’,) be the principal foci of 8, and §,, which 
cut the principal axis in A, B, respectively, so that AF,=/f, 
F’,=/',,..-. In Fig. 4 take the points X, (f,,/’,) referred to A 
as origin, and X, (f;, f’.) referred to B. Then since parallel rays on 
emergence from the system come from F,, F, is the image of F in §,. 
Therefore the line joining X, and F, on the y axis will cut the x 
axis in F. 
Again, since parallel rays on incidence go to F’, and thence to F’, 
F’ is the image of F’,in S,._ Therefore the line joining X, and F’, on 
the x axis will give F’ on the y axis. 
The principal foci of any system of surfaces may be determined 
in like manner. 
9. In the case of a lens the distances AF, BE’ may be readily 
found as follows in terms of fi; fi,... 
From the similar triangles FAF,, X,F’,F, (Fig. 4) we have 
AF TE eo AE. al 
Ae Tee: vee —e : 
where AB = e. 
Also from the similar triangles F’BF’,, X,F,k’, 
BF’ S.A BF’ Wan 
BF, ORF ie Si th—e 
These values can also be deduced from the relation of § 3. Thus, 
taking the x axis of the figure, 
BE Eo, 7 ae: 
10. In the system referred to in § 6 the lengths of the images 
({w;, w,...) which an object w at R, produces at R,, R,... may be 
determined as follows :— 
Let O, be the centre and f, /; the principal focal lengths of §,, 
&e. 
Then (Fig. 5) 
m -O;R, FX fi_ 4 
a ORB one ee 
In like manner, 
