DR. R A. SAMPSON ON A 



....... 



of vature i. a.wa ys alut two-fifths of the focal 

 IW S,'''o.,,,.lition for d-. of coma, .bid. ,ally ftiw as ABBB'S Sine Condition, 

 "^ to P1 " = Kl,r = KWJ-a,K = ,1,0-K' ; 



in-thb the right-haml member, apart from the foeal length,,, is a linear function of the 



quantities q. 



The condition fur absence of spherical aberration is 



= 



which is a quadratic function of q, ... . 



A numerical example of the use of such approximations will be given later. 



It is necessary to deal with express care with the case of the mirror. t may 1 

 treated as a single surface for which n = -1, and then 



^ = -2(3-e)B 8 , SJc-- -2B 2 , ^- = 0, 



A,/ = -SEP, .y = -23, .y = o, 



but this leaves the positive axis after reflection opposite to the direction of the ray. 

 It is better to reverse the direction of the axis, and this may best be done by 

 multiplying by the scheme {g, h; k,l} = {!,* ; *, -1}, and gives the following set 

 to represent the mirror : 



= 1, A=0, * = 2B, Z=+l, J>=-2B, 



i0 = 2B J , ^ = to = . M = 2B, V* = XJt = 0, 



i,Jt = 2(3-e)B, ^fc = 2B, ^Jfe = 0, 



SJ = 6B S , V = 2B, ^ = 0, ........... (10) 



the signs of all terms in k, I being reversed by this step, while g, h, \> remain 

 unchanged. Notice that the convention for the sign of B has not been altered, so 

 that, e.g., for the concave mirror B is negative, and the new value of k = (1 w)B is 

 negative also. 



If we write J,Jb = Jfc*+fcx+E, we must put x = it*. 



Besides the simple mirror I shall have also to deal with the system consisting of a 

 meniscus, silvered at the back. Such a system 1 shall call a lleverser. For neglected 

 thickness the coefficients follow readily from the case above (p. 35), of the juxta- 

 position <.f thmr thin lenses, replacing the middle lens by a mirror, and taking for 



