220 



duced from the laws of reflexion and refraction, however singular 

 it may appear that the latter laws should give any information re- 

 specting the former ; and he states that he has found this equation 

 to express rigorously a property of Fresnel's wave. His demon- 

 stration of this latter property having not yet been published, I 

 have been induced to investigate one for myself; and have thus 

 been conducted to a construction of the condition in question, so 

 simple that it may perhaps be mentioned here. Let r and w de- 

 note the planes vot and vop in the figure before referred to, which 

 may also be called the planes of ray-polarisation and of wave- 

 polarisation, for the ray ot, or for the corresponding wave ; and 

 let p', T, r', w' be analogous to p, t, r, w, but referred to any other 

 ray or wave ; then the following is the relation to be satisfied : 



OT . op', cos Rw' r= ot'. op . cos r'w ; 



Rw' and r'w denoting here diedral angles. Under this form, it is 

 easily proved that Fresnel's wave surface possesses rigorously the 

 property in question. Mr. Mac Cullagh's equation has been other- 

 wise obtained by M. Neumann, namely, as a condition for the pos- 

 sibility of depressing the equation of the vis viva to the first from 

 the second degree. 



On this and many other points of the investigation, Mr. 

 Mac Cullagh (as I have already said) has thrown out many inte- 

 resting and philosophical remarks ; for instance, that the perfect 

 adaptation which thus appears to exist between the laws of the pro- 

 pagation and those of the reflexion and refraction of light, is a 

 strong indication that these two sets of laws are derived from some 

 one common source, in other and more intimate laws not yet dis- 

 covered ; and that it is allowed to hope that the next step in phy- 

 sical optics will lead us to those higher and more elementary 

 principles by which the laws of reflexion and propagation are linked 

 together as parts of the same system. His remarks on the probable 

 connexion between the theories of metallic and crystalline reflexion, 

 and on the hopefulness of ascending to a true theory of light by 

 the method of mathematical induction from phenomena, (exempli- 

 fied, as has been seen, in his own papers,) rather than by attempting 

 prematurely to make deductions from dynamical principles, are 

 also well worthy of attention, though my own habits of thought 



