§64 THIRD REPORT — 1833. 



surfaces, and to two sets of caustic curves, and compose two 

 series of developable pencils, or ray surfaces ; so that each ray 

 of the final system may be considered as having, in general, two 

 foci, or points of intersection with other rays, indefinitely near. 

 The theorem here alluded to, namely, that of the general exist- 

 ence of two foci for each ray of a system proceeding from any 

 surface according to any law, was first discovered by Malus. 

 Mr. Hamilton also obtained it independently, but later, in 18^3. 

 It appears to be, as yet, but little known; but it is, he thinks, 

 essential to a correct view of the arrangement of rays in space, 

 for which the analogy of rays in a plane seems quite inade- 

 quate. Combining this theorem of the two foci with his view of 

 the characteristic function, and of the six constants of splierical 

 aberration, for the final system produced by oblique inci- 

 dence on an instrument of revolution, the author has found 

 that the two foci of a ray of this final system do not in 

 general close up into one, except for two principal rays, 

 having each its own principal focus. The interval between 

 the two foci of any other ray is proportional, very nearly, to the 

 product of the sines of its inclinations to the two principal rays; 

 and the tangent planes of the two developable pencils, passing 

 through any variable ray, bisect (very nearly) the two pairs of 

 supplemental dihedrate angles formed by the two planes which 

 contain this variable ray and are parallel to the two principal 

 rays ; in such a manner that all the rays of any developable 

 pencil of one set have (very nearly) one common sum, and all the 

 rays of any developable pencil of the other set have (very 

 nearly) one common difference, of inclinations to the same two 

 principal rays, or axes of the final system. These latter axes 

 always intersect each other, and their plane is either the 

 diametral plane of the instrument (containing the luminous 

 point or focus of incident rays), or a plane perpendicular 

 to that diametral plane, according to the sign of a cer- 

 tain quantity, which vanishes when the two axes happen to 

 coincide in one principal ray, round which the whole final 

 system has then a very perfect symmetry ; and, in general, the 

 angle of the two principal rays, whether in or out of the dia- 

 metral plane of the instrument, is bisected (very nearly) by a 

 certain intermediate ray in that plane, which may be called the 

 CENTRAL ray of the system, because the other final rays are 

 disposed about it with a certain symmetry of arrangement, 

 less perfect than the symmetry about an axis of revolution, but 

 resembling that of the normals to an ellipsoid about one of its 

 three axes, when unequal ; and accordingly the author finds 

 that the final rays from an instrument of revolution (when the 



