40 DR. R. A. SAMPSON ON A 



conjugate foci, P, P respectively, say at distances PO = u, OP' = v along the ray 



from tli- surface, so that 



u+v + kuv = 



where it IB to I* noted that the positive direction for both u and v is the direction 

 of the ray, which is reversed at the surface, so that if P, P' are found upon the same 

 side of the mirror u and t' will have the same sign we have the scheme 



g = l+hv, h = 0, k = k, 1=1 +ku, 

 \sitli tli.' roftHcifiita 



ie)]. (18) 



To obtain the system for a Cassegrain telescope, we must combine two systems, 

 (gfi...), (grT...), as in the Memoir, p. 160 (17), of which the former gives the great 

 mirror at its principal focus, by (17) above, while the latter gives the second mirror 

 between two conjugate foci, by (18). Let AC, e refer to the great mirror, and /, e' to 

 the second one. If we confine attention to spherical aberration, coma, curvature, and 

 astigmatism, it will suffice to form ^G, (5 a G, <$ ; ,G for the compound system, deriving 

 <J,H with the help of the equation (5 ; ,G ^H = H$. The resulting expressions are 



(5 3 G = -K'V/^I-K'U O + sVI + ieY'ttV [l -KuJ/K, (19) 



with 



The quantities e -l, e'-l are what SCHWARZSCHILD calls the deformations of the 

 mirrors, from spherical figures; when e = 0, or the deformation = -1, we have a 

 paraboloid ; if we choose them so as to annul coma and spherical aberration we have, 

 from the equations S 3 G = 0, J,G = respectively, 



while if we eliminate e from S 3 G, we get 



Curvature of field = -K 



= llu + K 'u{l+ K (-u+v)}/v, 

 and 



(20) 



