Mr. A. Smith on the Equation to FresnePs Wave-Surface, 335 



ture of the ascending and descending curves, and on the angle 

 at which the body ought to be projected, in order to return, 

 I must defer, as I have already trespassed too far on your 

 very valuable space. Should any of your readers misappre- 

 hend my meaning, or look upon anything I have said as re- 

 quiring explanation, I shall feel happy in affording any addi- 

 tional elucidation in my power. 



Belfast, Jan. 20. 1838. B. D. 



LI 1 1. Method of finding the Equation to FresnePs Wave- 

 Surface, By Archibald Smith, Esq,^ Fellow of Trinity 

 College^ Cambridge, 



To the Editors of the Philosophical Magazine and Journal, 



Gentlemen, 

 f WILLINGLY comply with the request of your Corre- 

 -■■ spondent, that I should communicate to your Magazine 

 the method of finding the equation to FresnePs wave-surface, 

 which was published in the 6th volume of the Cambridge 

 Transactions*. I shall do this as briefly as possible, indicating 

 the principal steps; the intermediate steps will, I believe, offer 

 no difficulty. 



If V be the length of the perpendicular from the origin on 

 a tangent plane of the wave- surface and Im n its direction- 

 cosines, a^, 6^, c^ the coefficients of elasticity ; the equation to 

 the tangent plane is 



Ix + my -\- nz =: V (1.) 



and we have the two relations 



Z2 + 7^2 + n^ = 1 (2.) 



V m" n^ __ 



^;2_a« + v^-b^ "^ v'^-c^ ■" ^ ••• ^^-^ 



To find the equation to the wave-surface we differentiate 

 these equations, making /, m, w, v vary. Eliminating the differ- 

 entials by the method of indeterminate multipliers, we get 

 the following equations : 



A 7 B^ 



^^ = ^^+ ^^2 (M 



y=.Am+^~j^^ (5.) 



Bw , , 



z = kn + ^_-^2 («.) 



-B.{(^.)%(^.)V(^.r} (7.) 



* See p. 78 and 261 of the present volume. 



