70 Mr. A. Whitwell on the Lengths of the 



where os— ~oc, and in front view by the line oV (fig. 10). 



The distance from the lens of the point s — ov l = v l . The 

 point Vi is conjugate to the point —u with respect to the 

 first lens. The ray as in passing through the second lens is 

 bent towards the axis of the second lens, the deviation *f or 



ae corresponding to the distance v x being =^v u since -j- 



is the deviation per unit length along the axis of a, this 

 deviation being measured along a line at right angles to the 

 axis od. The line af will therefore represent in end view 

 the ray after emerging from both lenses. The emergent 

 ray is represented in front elevation by the line o'fv 3 in fig. 10. 

 'Now v v 



os= j-oc= >' (h i cos0 l -\-h 2 sin Y ), 

 h h 



ae= ■^ad=-^(h 1 sin 2 + h 2 cos 2 ). 



From these values we can get as before the resolved 

 components of the diagonal af along the line oo' = k and at 

 right angles to tbis line =/. 



k= ^(/ij sin 2 2 + h 2 sin 2 cos 2 ) ~~ ir (]h cos 2 #i -r ^2 sm #i cos X ) + hi, 



l-=.-±(h 2 cos 2 2 + h } sin 2 cos 2 ) — j- (1i 2 sin 2 6 i -f A x sin X cos X ) + h 2 . 

 h J\ 



The corresponding values m and n for the resolved com- 

 ponents of the line bj, obtained by considering a symmetrical 

 ray incident at b and making a similar construction, are 



m= y (Aj sin 2 2 — h 2 sin 2 cos 2 ) - ~ {h Y cos 2 L — h 2 sin 2 cos } ) + h ly 



n= ir(h>2 cos2 ^2~^i s i n ^2 cos # 2 ) — -r (Jh sm2 #i + \ sm #i cos ^) + h 2 . 

 h /1 



By equating Horn and Z to n we get the condition under 

 which the lines af and bj intersect on the line oo\ viz., 



/j sin 2 cos 2 —f 2 sin 2 cos ^ = 0. . . . (6) 



If the semi-length of the axial focal line = Z,, from 

 figs. 9 and 10 we have 



h 2 ~z, i; 



or - 7 &■' 



li = hi-.jti2. 



