336 Mr. A. Smith 07i the Equation to Fresnel's Wave'Surface, 



From these seven equations the six quantities /, m, n, v, A, B 

 are to be eliminated, and the resulting equation will be that 

 to the wave- surface. 



Such an elimination would in general be quite impracti- 

 cable ; in the present instance, and in many others of the same 

 class, it is facilitated in a remarkable manner by the forms of 

 the equations. 



The equations (4.), (5.), and (6.) multiplied by /, 7W, n re- 

 spectively and added, give 



w = A (8.) 



The same equations squared and added, give 



T> 



If we put r^ for x'^ + 2/^ + ^^ j and for A the value just 

 found, we shall find 



B = w (r*— i;^) (9.) 



If these values of A and B be substituted in equation (4'.) 

 it may easily be put under the form 



— (Ir^^x v^). 



The same substitution made in (5.) and (6.) will give two 

 similar equations ; and if these three equations be multiplied 

 by ^5 2/, z respectively, and added, the right-hand side will be 

 found equal to zero; and thus we get finally 



as the equation to the wave-surface. 



There is another form of the equation which is rather more 

 easily obtained than the above, which will be found in a paper 

 inserted by Mr. Gregory in the first Number of the Cambridge 

 Mathematical Journal. The form I have given has this ad- 

 vantage, that we may derive from it immediately the construc- 

 tion by means of the ellipsoid ; for this I may refer to an 

 article in the second number of the Journal just mentioned. 



Jordanhill, near Glasgow, Arch. Smith. 



March 6, 1838. 



