Formula and Methods connected with Lenses. 449 



(about 3 sq. mm. in area) is placed in the centre of the 

 upper or convex side, and the tubes of the telescope, still 

 standing on the plane mirror, are manoeuvred until the image 

 of this paper coincides with itself. The length is then found to 

 be 25*15 cm. The focal length of this lens can then be 

 determined as follows: — 



The principal focus of the lower lens is above the lower 

 surface of the upper lens by the amount of the shortening of 

 the tubes, i. e. 25*95— 25*15, or '80. But the position of the 

 second principal focus of the upper lens is 19*92 below this 

 face. Therefore the value of k is -(19*92 + -80)= -20*7-2, 

 the negative value being the result of overlap in regard to 

 the two systems. 



But the amount by which the outer principal focus moves 

 upwards is —21*12. 



Hence /, 2 



-20*72 



21*12. 



Hence f 1 = —20*92, the negative sign being of course chosen. 



Again, if a convex lens, whether achromatic or not, is laid 

 upon a plane mirror, its convex side downwards, a few drops 

 of any refracting liquid being placed between the two, the 

 principal focus will travel a distance depending on the index 

 of refraction of the liquid. Care must be taken that the 

 combination is convex in character. 



If r is the radius of the face common to the lens and the 

 liquid and p is the index of refraction of the liquid, the 



r 



liquid lens will have for focal length — — r, and its first 



r 



principal focus will be situated at =- below the plane 



//, — 1 



mirror. If v be taken as the distance from the wetted surface 



of the lens of its own second principal focus, the k of this 



problem is clearly \ — -y — v j- 



Hence if/ be the focal length of the lens, and c be the 

 travel of the first principal focus due to the insertion of the 

 liquid, 



or 



Iff 2 



r 



v 



[(P \ 1 



Phil. Mag. S. 5. Vol. 49. No. 300. May 1900. 2 I 



