Lengths of the Focal Lines of Cylindrical Lenses. 59 



Using the data of the above experiments, if i=45°, iV=l, 



f c =f t =D (nearly) = 23 em., cos 0' — cos i = '2264, 



£=•9 cm., A, = 60xl0- 6 , 



d0'/fl?n=4?3 xl0~ 6 , 



whence 



7 = 146 x 10~ 6 radian, 



or about a half minute of arc per fringe, and /3 = 44' per 

 fringe. Thus j3 is about 88 times as large as 7. At i— 22 0, 5, 

 Y = l''5 per fringe, y3 = 45' per fringe. Naturally the sensi- 

 tiveness increases with the angle of incidence i. When the 

 fringes are large 1/10 fringe is easily estimated, so that a 

 horizontal angle 7 of a few seconds between mirror and 

 grating should be measurable. An ocular micrometer as 

 suggested would carry the precision beyond this. 



Brown University, 

 Providence, R.I. 





£=. ■ 



IV. On the Lengths of the Focal Lines of Cylindrical Lenses. 

 By A. Whitwell, M.A., A.R.C.Sc.L* 



HPHE following paper is a continuation of one entitled 

 JL " On Refraction at a Cylindrical Surface," published 

 in the Phil. Mag. for July 1903 |- In that paper the form of 

 the focal lines or focal areas produced by refraction at a 

 cylindrical surface was investigated, the aperture parallel to 

 the axis of the cylinder being considered to be unlimited. 



The object of the present paper is to find the lengths of 

 the focal lines produced by a single cylindrical surface or by 

 one or more cylindrical or sphero-cylindrical lenses, the 

 aperture being so small that the focal lines may be con- 

 sidered to be straight lines. The formulae arrived at are 

 analogous to the ordinary first-approximation formulae for 

 thin spherical lenses. 



* Communicated by the Author. 



t There is an error in this paper on p. 54. The two equations at the 

 foot of this page should be 



< 



i 



r 



d+y/ 



'tf + tf' 



~ d" 



H 2 (a - 



r-+(/* 2 



-l)h 2 



■{a-r\ 



