82 Mr. E. Howard Smart on a Formula for 



the tangential semi-aperture, and /i the focal length of the 

 piano-cylindrical component of the lens. The vertical image 

 bands formed at the tangential focus will have a horizontal 

 width 



wv 3 2h 2 v z 



the first term being the width the bands would have were 

 the lens spherical. The distances between the centres of the 



vertical image bands at the tangential focus will be 



^3 



— u 



If fig. 17 be regarded as a plan view analogous to fig. 16 

 instead of an elevation analogous to fig. 14, then all that has 

 been said about the images of the horizontal bands at the 

 axial focus will apply to the images of the vertical bands at 

 the tangential focus. The lines A to I are, however, to be 

 obtained by putting a=l, 2, 3, or 4 in the formula 



a f\U 



«/i±2/> 2 /2* 



V. A Formula for the Spherical Aberration in a Lens- 

 System correct to tJie Fourth Power of the Aperture. By 

 E. Howard Smart, M.A., Head of Mathematical Depart- 

 ment, Birkbeck College*. 



I^HE ordinary formula? as given in the text-books for 

 central spherical aberration are computed to the square 

 of the aperture only, — a degree of approximation which is 

 insufficient for the purpose of the practical optician in the 

 design of photographic and other objectives. In the 

 following work a formula will be given for the longitudinal 

 aberration for a system of coaxial spherical surfaces separating 

 media of refractive indices fi ...fjL K which is correct to the 

 fourth power of the aperture. A greater degree of accuracy 

 than this is usually undesirable, the complexity of the 

 additional corrections being out of all proportion to their 

 usefulness. 



Let the spherical surface AP of radius n separate media 

 of refractive indices /U{_i and fii. Let the ray OP be incident 



* Communicated bv the Author. 



