79 



culating focal crystals; the directions of osculation are the directions of the lines of extraordi- 

 nary/ refraction, analogous to those lines of ordinary reflexion and refraction which were before 

 considered : the caustic surfaces of the extraordinary system contain also the centres of the 

 greatest and least spheroids which osculate to the surfaces of constant action ; the intersection of 

 the caustic surfaces reduces itself in general to a finite number of principal foci, analogous to the 

 principal foci of ordinary systems, considered in the two first parts; each principal focus is the 

 centre of a spheroid which has contact of the second order with a surface of constant action 5 it 

 is also the focus of a focal crystal which has contact of the second order with the surface of the 

 given crystal ; tlie partial differential equations which represent crystals corresponding to a given 

 caustic surface, are to be integrated after the manner of the second part, . 154, 155. 



Condition for the rectangularity of an extraordinary system, expressed by a partial differential 

 equation of the second order ; integration of this equation ; the integral expresses that the nor- 

 mals to a surface of constant action are tangents to a cylindric surface, whose generating line is 

 parallel to the axis of the crystal, , . . . 156. 



The preceding results may be extended to the extraordinary systems produced by reflexion at 

 the interior surface of the crystal ; when the extraordinary rays recover their ordinary velocity they 

 become again perpendicular to the surfaces of constant action : this theorem enables us to apply 

 the results of the two first parts, to systems produced by combinations of crystals, mirrors, and 

 lenses, , , . , , . 157. 



XXIX. On other extraordinary systems. 



Law of extraordinary refraction in crystals with two axes ; characteristic function of a system 

 produced by such a crystal ; spheroids of Brewster ; surfaces of constant action ; the results of 

 the preceding sections may be extended to these systems, . . 158,159. 



Remarks on systems produced by crystallized mediums of continually varying nature, 160t 



XXX. Lavi of least action. 



General expressions of this law ; development of these expressions by means of the calculus of 

 variations. In every optical system, the action may be considered as a characteristic fukc- 

 TioN, from the form of which function rnay be deduced all the other properties of the system. 

 This function (when we know the luminous point, and the reflecting or refracting media) de- 

 pends only on the coordinates and on the colour ; its partial differentials, of the first order, taken 

 with respect to the coordinates, are in ordinary systems of the form 



Jj U Sj 



-=«.«. .^=«.^. -j^=«.y. 



and in extraordinary systems of the form 



VOL. XV. N 



