74 Mr. A. Whitwell on the Length* of the 



From fig. 12 we have 



lo r 4- 8 — v. 



/<2~ V! l ' 



or 



/ 2 =- 2 (r 2 4-8)-// 2 (12) 



' ow -i+i=4 (13) 



M *>1 /l 



and 



1 11 



+ T = ,T (1*) 



,\ 1_ __ ^2—y«~ ^2/2 + ''2/1 -/i/a 



t'l s%/i— S/1/2—V2/1/2 



and substituting this value in (12) we get 



L _ M'y(/ 2 - /i-g) + r 2 (28/-, -8^) + sy 2 } . 



If r,=f,, 



From fig. 11 it will be seen that when c f d' = c'vi\ we shall 

 have r 1 = r 2 4- 8 and the lengths of the focal lines =0. 



This will happen when the points conjugate to —u with 

 respect to the two lenses separately coincide, or when 



*i'(fi-fx-t} +r 1 (5 2 + 2S/ ] )-5 2 /i = 0. 

 The form of this equation shows that when f and/ 2 are 

 constant there are two values of t^ corresponding to each 

 value of 8 which will make the lengths of the focal lines 

 = 0. 



7. To find the lengths of the focal lines of a sphero- 

 cylindrical lens when the source of light is not on the 

 optic axis. 



(a) The axial focal line. 



Fig. 13 is a front elevation of the system. We can regard 

 the lens as made up of a piano-cylindrical lens of focal 

 length f and of a piano-spherical lens of focal length / 2 . 



The two powers of the combination are -? and ~y -f -r, the 



1 >> 2 J 1 / 2 . 



latter of which we will call -p. Let a be the source of light 



situated at a distance -an = a from the axis ow and at a 



