Puimmkr — On the Symmetrical Optical Instrument. 77 



Here the sign has been changed throughout, the result of reversing the 

 sign of ?•,;. This has been done because the reflecting surfaces are usually 

 concave to the incident ray. With this change of sign 



T, » l 1 l 



or, if A ; is the vertex of the mirror and C ; . v C; the points where the 

 conjugate planes meet the axis, 



K t = 1/C^Ai - 1/n = 1/n - 1/C'iAi. 



After the reflexion it is perhaps most convenient to reverse the axis of x at 

 the vertex ?•,-• Let d ; . be the (positive) distance between the mirrors i and 

 i + 1. Then, c,-, c{ having double meanings for the two directions of the 

 axis, 



d; = C;A; n + C;Ai = («,- +1 - C() + (rt,; - C{) 



= G/A !n + Of A; = (a in - c{) + {at - c{). 

 The necessary changes in the remaining formulae give 



— — '-- - X 



«i - cj'-! n n «, : - a" 



- k; = Zp t f 1, - K[' = K in - 1, 



- L{ = Lan + 1, - L" = L t v, - 1, 



h ln _ r in Lffj - 1 h{ n _ r,- n K#\ - 1 



; -Ei+i'ni + 1' hi 1'; ' A'- +1 r,- +1 + 1' 



h = ri - h' = ^ r i±i ajl' = 



1 7" „. i 1' 1 ». / T W\' ' ' 



z^ + r J r, (Zj - zi)' ' ' z f - /r/ 



-"itl _ -"-I'+i'V+i + 1 -"/ +i _ A'n?7+i + 1 



#< = Kn-t-1 ' Hi Ltn - 1 



No further alteration is required. Schwarzschild has given a treatment of 

 the mirror system which is independent of the previous theory for a refracting 

 system. 



BJ.A, l'ROC, VOL. XXXIV, SECT. A. [18] 



