Lord Rayleigh's Investigations in Optics. 

 Fig. 11. 



481 



D A 



approximately. For glass /a — 1 = J nearly; so that the old 

 rule*, that " in glass lenses the half-breadth is a mean propor- 

 tional between the thickness and the focal length," is more 

 scientific than the usual formula in terms of the radii of cur- 

 vature. If the lens do not terminate in a sharp edge, we may 

 take as the effective thickness the difference of the thicknesses 

 at the centre and at the edge. 



For an oblique central pencil, the focal lengths in the two 

 principal planes may be obtained as in the case of the mirror. 

 They take the form 



y' 2 cos 2 (/> _ 



2/1 



(jul COS <f/ — COS <j))t, 



tf - 



&F=(/^ COS <j)' — COS <j))t, 



Aj2 



(9) 

 (10) 



in which, if we please, we may substitute for t its value in 

 terms of the radii of curvature, viz. 



l ?y 2 

 2U 



\r s/ 



§ 6. The Aberration of Oblique Pencils. 



In treatises on geometrical optics it is usual to calculate the 

 aberrations of mirrors and lenses for direct pencils, but in the 

 case of oblique pencils to rest satisfied with determining the 

 primary and secondary focal lengths. For most purposes 

 indeed astigmatism is a worse defect than aberration, so that 

 in the presence of the former it is not worth while to consider 

 the latter ; but in this respect the spectroscope is an exception, 

 and the completion of its theory requires the consideration of 

 the aberration of oblique pencils. 



The reason of this peculiarity is not difficult to see. When 

 a luminous point is observed through an optical instrument 

 affected with astigmatism, there are three notable representa- 



Cocldingtou's 'Optics,' p. 96. 



