CASSEGRAIN REFLECTOR WITH CORRECTED FIELD. 31 



it will be free from spherical al>erration and from coma, and the images of points will 

 be circles in the plane through the principal focus, the radii of which are given by 

 1031 3.3" x (a //') x /S^H. If <J,H, which by (2) is made equal to -S 3 G, is not zero, the 

 instrument will be successful for such values of the angular radius of the field as keep 

 this down below desired limits. These conditions give the objects which I aim at 

 attaining. Given the general design of the instrument as regards apertures and focal 

 lengths, it will be found that the lens which is used as a mirror, or the Reverser as I 

 shall call it, is completely determined in its curvatures by the conditions for 

 achromatism, and the quantities available for adjustment are the figure of the great 

 mirror and the curvatures of the two lenses of the corrector. These are used to 

 satisfy rigorously equations (2), and the essential difficulty of the problem is to find a 

 case among the great number of those that- are open for trial, the solution of which 

 shall prove to be of a practical kind, not involving excessive curvatures. Once an 

 approximate solution is obtained, to refine it only requires patience, but to arrive in 

 the neighbourhood of a solution is a problem in which trial needs some guide. In 

 this connection I would draw attention to the theory given below of the Thin 

 Corrector. This is an optical system of two or more thin lenses in contact, null as 

 far as deviation and colour are concerned, and introducing aberrations only which are 

 available for correcting existing aberrations. Thus simplified, it is manageable 

 algebraically, and its indications will show the possibility or otherwise of any projected 

 arrangement. 



If we denote by 3) the curvature of the field and by $ PETZVAL'S expression 



being the curvature of the surface (2r), as in the Memoir, p. 162, we have 



S a 



at the principal focus ; hence <$ a H which gives the amount of astigmatism is 

 determined by 



.......... (3) 



a result which can also be deduced at sight from known expressions for astigmatism 

 and curvature of field according to SEIDEL'S theory. In the special case of a flat field, 



or 9J = 0, it becomes 



........ . (3A) 



and this may be taken in place of the third of equations (2) as one of our necessary 

 conditions. We notice that it is only possible to control the astigmatism through the 

 value of *JJ, and the value of $ depends only in small degree upon the distribution of 

 curvatures between the two faces of a lens. It is a matter then of the general design 

 of the instrument to keep S a G down to a suitable magnitude. This presents no 

 difficulty. I have been content to keep it small enough for my purpose. If a field of 



