366 THIRD REPORT — 18^3. 



of this theory. A new track seems to be opened thus to ma- 

 thematical and practical opticians. 



The principle of the characteristic function, from which have 

 been deduced the foregoing results, among others not yet pub- 

 lished, respecting optical instruments of revolution, may be 

 applied to every part of mathematical and perhaps of physical 

 optics ; and an analogous function and method may be intro- 

 duced in other sciences, especially in dynamical astronomy*. 

 But the author confines himself to mentioning the application 

 which he has made of the principle to the study of the laws of 

 extraordinary refraction in the crystals called biaxal. The 

 general laws of reflection and refraction, ordinary and extraor- 

 dinary, at any point of any surface, are expressed by his 

 function as follows, when the normal to the reflecting or 

 refracting surface at the point of incidence is taken for the 

 axis of z : 



J-^^0;J^ = 0: (8.) 



and in the language of the undulatory theory they may be 

 enunciated by saying, that if the normal slowness of propa- 

 gation of a luminous wave, at any point of incidence on any 

 reflecting or refracting surface, be decomposed in any direction 

 parallel to this surface at this point, the component of normal 

 slowness is not altered by reflection or refraction. In the case 

 of ordinary refraction, this comes to saying, that if on the in- 

 cident ray prolonged, and on the refracted ray, we measure 

 from the point of incidence lengths represented by the indices 

 of the first and second media, those lengths will have one 

 common projection on the refracting surface or on its tangent 

 plane ; which is a form for the law of Snellius. For extra- 

 ordinary refraction, we must in general construct the normal 

 slowness of a wave by a variable length not always coinciding 

 with the ray ; but the two lengths thus substituted for the two 

 successive indices will still have one common projection on the 

 refracting face of the crystal, if plane, or on its tangent plane, if 

 it be curved. If now we seek the locus of the end of the line, 

 which represents in length and direction the normal slowness 

 of a wave, for all possible directions of this slowness, we get for 

 ordinary media a sphere, but for extraordinary media (on 

 Fresnel's principles) a certain double surface, which is not the 



* See the Dublin University Review {or October 1833. Mr. Hamilton has 

 since developed the dynamical application of his principle, in an essay On a 

 General Method in Dynamics, which has been presented to the Royal Society, 

 and ordered to appear in the Philosophical Transactions (or 1834. 



