334 A. TANAKADATB 



multiplying (2) by ß, and adding (1) (2) (3) we get 

 1^1^ 2(^-1) ^ 2// 



Pi Pi n 9\ 



This shows that tlie lens is equivalent to a mirror whose curvature is 



equivalent to — (>a — l)/i\ + u/r.,, and is concave or convex according 



as {u — l)/;'i < or > ///r^ . This equivalent mirror becomes a plane 



wdien i]/r^ ■=-- {ß — l)/u . 



Let — be the curvature of this equivalent mirror, or as we shall 

 Pi _ . , 



call it for brevity, the " equivalent curvature," then 



1 u-i . u 



(4) 



Now reverse tlie lens, that is interchange the reflecting and re- 

 fracting faces, and let the equivalent curvature be — then 



P-2 



1 _ /^-i u 



pi r., 1\ 



But the principal focal length of the lens is given by 



^^"'-K^-i) 



From (4) (5) (ß) we get at once * 



(5) 



(6) 



1 1 



P^ f 



1 _J_ J_ 







1 











~T 







1 



/'l 



+ 



I 



77 



+ 



■> 



} 



(7) 



Tlip sifïiis of j-j and vo are eonsidered with repaid to the first position of the lens. 



