CASSEGRAIN REFLECTOR WITH CORRECTED FIELD. 39 



which give <$,/, t\h' ; and similarly we have $ } tf, <?,/'. Form also 

 -/<M,G -I- (gl + f,k) S,G-gk<\G = ...+ 



-hlS t 1 1 t . . . 

 and 



AM.H-... 



\\ itli similar equations in it.K, iT.L. These equations, for example, answer the question 

 df what al>errations are shown when a known system is reversed and presented with 

 the opposite face to the beam, the unit-points Iwing simply interchanged so that the 

 normal effect as shown in the position of the focus is the same as before. For if an 

 unaherrant heam originating ;it < is Drought to a focus at ()' and shows there 

 aberration ooaffioienta <\y, ... ; or, what is the same statement, an alwrmnt )>eain with 

 coefficients S t g, ... emerging from (V and passing through the system in the opposite 

 direction is brought to an unaberrant state at O, then if S t g f , ... are the coefficients 

 introduced by the reversed passage we have the joint effect of A,^, ... superposed to 

 \{j, ... is null, or <^,G, ... are all zero. But it must lie noted, as was pointed out for 

 the mirror, that as the direction of the axis is reversed the signs of c$,...<y must be 

 reversed before they are brought into the equations with <?,0r', ... ; further, since 

 G = 1, H = 0, K = 0, L = 1, we have g f = /, It' = -li, k' = -k, I' = g, and n = 1. 

 The whole question has some general interest, but I shall not pursue it further at 

 present, because it is somewhat beside our mark, and I return to considerations that 

 bear upon the main problem. 



Coming now to the immediate object of my paper, which is the Cassegrain 

 telescope, I shall first consider what can be effected with two mirrors simply, which 

 will give opportunities for writing down useful expressions of various forms relating 

 to mirrors. 



A mirror with both origins at its surface, and the reversal included, gives the 

 scheme (10) p. 36, or say 



g = 1, h = 0, k = k, 1=1, p = k, 

 where k = 2B, together with the aberration coefficients 



#, 0, 0; *, 0, 0; i(2 + e)i 3 , &*, ; $k>, k t ..... (16) 

 With the surface for one origin and the principal focus for the other, these become 



g = o, h = -k~ l , k = k, 1=1, 

 with the coefficients 



-bk 3 , -, 0; -i*. -1, 0; ibid. ; ibid ..... (17) 

 It by the formulae of the Memoir, p. 164 (22), we transfer the origins to two 



