TRANSACTIONS OP THE SECTIONS. 367: 



same as Fresnel's curved wave-surface, propagated in all di- 

 rections from a point, but is connected therewith by several 

 remarkable relations of reciprocity, and may be called the 

 surface of components, since its coordinates are themselves the 

 components of normal slowness of propagation. They are 

 equal to the partial differential coefficients of the first order 

 of the author's characteristic function V, and are connected 

 by a partial differential equation of the form 



which may be regarded as the equation of the surface. And 

 the general equations of reflection or refraction (8.), when put 

 under the form 



8V . JV 8V 8V^ .8V_8V 



y^ + ^^ = -87' ^ + ^87-T^' ^^^'\ 



express that the corresponding points on the two surfaces of 

 components, before and after any reflection or refraction, or- 

 dinary or extraordinary, are on one common ordinate to the 

 reflecting or refracting surface, or to its tangent plane ; which 

 gives a new and general construction for the direction of a 

 reflected or refracted wave, and therefore for that of a re- 

 flected or refracted ray, simpler in many cases than the con- 

 struction proposed by Huygens. Thus, if it were required to 

 determine by this new construction the direction and the un- 

 dulatory velocity of an extraordinary ray, refracted in Iceland 

 spar, being given the direction of the incident ray in air, we 

 should have to construct first the two successive surfaces of 

 components, which would be here a sphere for the air, and a 

 spheroid (not the Huygenian) for the crystal, the common 

 centre of both being at the point of incidence ; and then, after 

 determining the point of the hemispheroid within the crystal, 

 which is on the same ordinate to the refracting face as the 

 point where the incident I'ay prolonged meets its own interior 

 hemisphere, we should only have to draw a tangent plane to 

 the spheroid at the point thus determined, and to let fall a per- 

 pendicular on this plane from the point of incidence ; for this 

 perpendicular is, in length and direction, the radius vector of 

 the Huygenian spheroid, and therefore represents the undu- 

 latory velocity and the direction of the extraordinary ray. And 

 other more complicated cases may be treated in a similar 

 manner, either by using a construction of this kind, or by the 

 equivalent formulae derived from the characteristic function. 

 When the author proceeded to apply this general method to 



