Lord Rayleigh's Investigations in Optics. 43 



For some purposes a more convenient expression may be 

 obtained by substituting for v its approximate value. Writing 



/// for "-—, we get 



G 



i 2 _f =( ^_ c) (i_i)ri + 3^((v±i) £+ ^ 



u v v 7 V s/L zfic I u s 



+ ^!±i}]. . (12) 



If tbe incident rays be parallel, u = co , and the aberration 

 vanishes when 



* = ^ = /" 2 cos 2 <£' / 13 n 



/• /// — /j/ 2 + 1 /x cos $' cos <£ — ^ COS 2 (// + COS 2 (/>' ^ ' 



If the second surface be plane (s = oo ) ? this condition is 



fi'-fj,' 2 + 1=0, 

 or 



^=i(l+V5) = l-62. 



For a refractive index //,= 1*5, the value of <p which makes 

 jj/= 1*62 is about 29°. This is the obliquity at which a plano- 

 convex lens of plate glass must be held in order to give a thin 

 primary image. A more refractive lens must be held at a less 

 obliquity. If the refractive index exceed 1*62, there is no 

 position of the lens for which the image is free from aberration. 



From the above formulas there is no difficulty in calculating 

 the aberration due to any combination of lenses. As an ex- 

 ample I will take the case of two lenses of equal focal length, 

 inclined at equal angles but in opposite directions to the axis 

 of the pencil. Denoting the radii of the second lens by v' 

 and /, the final focal length v r is to be found from (11) in 

 combination with 



+(^+«)(i + i)}J. 



By addition, 



«2 „2 



-^-.>(H)& + W{<^>(H) 



+ 0«'+«)(!+-;-i-i)}]. • («) 



If the lenses be of the same curvatures and be similarly turned, 



