30 DR. R A. SAltfPSON ON A 



Tin- iiictliinls which I employ are those of a memoir recently published.* 

 SCHWARZSCHILD usetl the Characteristic Function. Our methods thus differ, but 

 since aberrations of the third or any other order are the same things, no matter how 

 they are obtained, where we occasionally touch the same matter the differences are 

 at most those of notation, and occasionally these are slight ones. I have not 

 attempted to remove them because it seems to me that an investigation is easiest 

 to read if expressed in notation that grows naturally out of its own processes. I 

 shall therefore adhere strictly to the notation of my Memoir, amplifying its results 

 88 occasion requires. 



We may take for reference the following specifications of the faults of an optical 

 field at its principal focus in terms of the coefficients <?,G, &c. : 



a = semi-aperture. 

 f = effective focal length. 



/3 = tangent of inclination of original ray to axis. 



Position of least circle of spherical aberration . . . Sf = + f / V^G. 

 Angular radius of this circle ........ 25783" x ^- xa*K G. 



t/ 



Comatic radius ............ 1 03133" x x /W,G. 



Secondary focal line after principal focus .... 



Primary focal line after secondary ..... f'ffS. H 



Radius of focal circle ..... 103133"x ^ 



Curvature of field (convex to ray if positive) . . . (l/^ 

 Distortional displacement ...... 103133"x ( 1 //') x 



...... (1) 



With respect to these it may be explained that the Comatic Radius is the radius of 

 > around which rays from a zone of radius a are distributed, the centre of the 

 circle being displaced from the normal image-point by an amount equal to its 

 the "secondary" focal line is the line in the plane of the axis; the word 

 ana after, in the order in which light reaches the points ; the focal circle 

 cle half way between the two focal lines, through which, in the absence of 

 the zone would pass ; the curvature of the field refers to the field 

 the focal circles of all object-points. 

 Now, if we secure a field for which 



= 0, ...... (2) 



"A New Treatment of Optical Aberrations," 'Phil. Trans.,' vol. 212, pp. 149-185. 



