78 



Aberrations in an undevelopable system ; virtual developments of the pencil ] 34, 135. 



XXVI. On systems of the second class. 



General formulae for these systems; condition of rectangularity ; equation of the surfaces 

 which cut the rays perpendicularly when this condition is satisfied . . 136. 



Pencils of the system ; caustic surfaces ; when the system is rectangular, the intersection of 

 these surfaces reduces itself to a finite number of points . . 137,138. 



Virtual foci of a ray ; virtual caustic surfaces ; diametral surface; principal virtual foci 139. 



Law of the variation of the virtual focus ; the planes of extreme virtual foci cut one another at 

 right angles ; generalization of the results of the IXth section respecting the properties of thin 

 pencils . . . . . 140. 



On emanating systems .... . 141. 



Foci by projection ; the planes corresponding to the extreme foci by projection, coincide with 

 the planes of extreme virtual foci ; they furnish a pair of natural coordinates, which are of exten- 

 sive use in optics, ...... 142. 



Osculating focal surfaces ; the greatest and least have their foci upon the caustic surfaces, and 

 osculate in the directions in which the developable pencils intersect the surface from which the 

 rays proceed ...... 143. 



Applications of this theory . . . 144, 145, 146, 147. 



Caustics of a given curve ; conditions of integrability, which are necessary for the existence of 

 focal surfaces . ... . 148. 



XXVII. On syitems of the third class. 



Object of this section ; limiting surface enveloped by all the cones and other pencils of the sys- 

 tem ; condition for this surface being touched by all the rays ; remarks on the investigations of 

 Malus, respecting systems of this kind . . 149, ISO, 151. 



XXVIII. On extraordinary systems produced by single-axed crystals 



Object of this section ; analytic expression of the law of Huygens; principle of least action; 

 characteristic function of an extraordinary system . . 152, 153. 



The surfaces of constant action are touched by spheroids, having their centres on the extraordi- 

 nary rays, and the rays may be considered as proceeding from these surfaces, according to a simple 

 lav/ ; when the extraordinary rays converge to one focus, the surfaces of constant action become 

 a series of concentric spheroids, and it is always possible to assign such a form to the surface of the 

 crystal as to satisfy this condition : hence it follows, by XXVI, that the extraordinary rays are 

 in general tangents to two caustic surfaces, which contain the foci of the greatest and least os- 



