68 Mr. A. Whitwell on the Lengths of the 



By equating k and m or I and n we get the relation 



f 1 sin # 2 cos 6 2 —f 2 sin 0j cos M 



and it will be found that this relation also satisfies the 

 equation P + Z 2 — ??i 2 4- ^ 2 . 



If this relation between Q x and 6 2 hold, then every pair of 

 rays incident at symmetrical points such as a and h will, after 

 refraction by both lenses, intersect in the central plane at 

 some point on a line represented in iig. 7 by op, and in fig. 8 

 by pq. This line pq is therefore the principal focal line. 



If we call its semi-length l im 



From fig. 7 we have l l = op = h l — o'p. 



Now o'p __k j _ , ' k 



Substituting the known values of k and I with the condition 



fi sin 6 2 cos 6 2 -f 2 sin X cos 6 U 

 and simplifying, we get 



/, cos 20s +/ 2 cos 20! 



li-*i 



/i COS 2 2 + /' 2 COS 2 6 X 



If Fj be the principal axial focal length of the combination, 

 we have 



* *' /i"T' 01 ^^I-T"—' 



and substituting the value of Z with the condition 



/i sin 6 2 cos 02= f 2 sin ^l cos 1? 

 we have 



F,= 



/i cos 2 2 +/. COS 2 0, COS 2 6 S COS 2 ^ ' 



, , „ /cos 20, , cos 26>A 



(2) The tangential line. 



If we produce the line af (fig. 7) to meet the axis of e in r, 

 then or will be the semi-length of the tangential focal 

 line. If or = l 2 we have from fig. 7, 



l 2 _ or __ A 2 _ / 

 /j. op o'p k' 



