TRANSACTIONS OF THK SECTIONS. SG/i 



incident rays are oblique) are very nearly normals to a portion of 

 such an ellipsoid, having the central ray for one of its three un- 

 equal axes, and having the two principal rays for its two umbili- 

 cal normals, at two out of the four points where the ellipsoid has 

 complete contact of the second order with an osculating sphere. 

 The centres of the two osculating spheres at these two points 

 are the two principal foci of the system; and the centres of the 

 two extreme osculating spheres at any other point of the ellipsoid 

 are the two foci of the corresponding ray, or the points at which 

 that ray touches the two caustic surfaces. These latter sur- 

 faces are, in the present approximation, the surfaces of centres 

 of curvature of the elhpsoid : they have a curve of intersectioa 

 with each other, which contains the two principal foci ; every 

 point upon the curve, except these two, being the first focus of 

 one ray and the second focus of another. A plane may be 

 drawn perpendicular to the central ray, and passing through 

 the two principal foci ; and this plane will cut the two caustic 

 surfaces in sections which compose a kind of little lozenge, con- 

 sisting (very nearly) of two curvilinear equilateral triangles, 

 having the principal foci for two common corners : the quadra- 

 ture of these curvilinear triangles, and of the other sections 

 of the caustic surfaces, depending on elliptic integrals. In all 

 the foregoing remarks, it is supposed, for greater generality, 

 that the aberrations do not vanish with the obHquity of the 

 incident rays ; but when the instrument is aplanatic for direct 

 incident rays, it is easy to apply the same theory of the charac- 

 teristic function and the six radical constants of aberration, 

 and to determine, for this particular case, the components of 

 spherical aberration which arise from obliquity only. 



This theory of the aberrations of oblique rays, for an optical 

 instrument of revolution, may admit of practical applications. 

 For the mathematical symmetry of arrangement of the final 

 rays about the central ray of their system, and the intensity of 

 the two principal foci, may perhaps affect our sight, and have 

 some appreciable influence on the practical performance of an 

 instrument ; but of this Mr. Hamilton speaks with diffidence, 

 because experiments directed expressly to the question appear 

 to be required for its decision. If the mathematical properties 

 which he has determined by theory in the arrangement and 

 aberrations of a system, shall be found in practice to have any 

 sensible influence on the phaenomena of oblique vision, it will 

 become necessary to alter some of the received rules for the 

 construction of telescopes and microscopes ; or, at least, it will 

 be possible to improve those rules by following the indications 



