86 Mr smith, ON THE INVESTIGATION OF THE EQUATION 



of Fresnel and Ampere, and state shortly the steps by whicli they are 

 obtained. 



In the Memoir of Fresnel referred to above it is shewn, that if 

 a section of the " surface of elasticity", whose equation is 



(x- +if + z'Y = (rx-+b^i/° + (fx:^ (1 ) 



be made by the plane 



z = »i.r + fii/ (2), 



the greatest and least radii vectores of the section will be the values 

 of V, derived from the equation 



(«- - v") {r - V-) if + (b' - v') (c- - V') nf + («- - v') (Jr - v') = . . . (3), 



and that if a plane be taken, parallel to the former and wliose distance 

 from the origin is one of the values of v, this plane, whose equation is 



z = })ix + 7i!/ + V \/l + ni' + tf (4) 



will be a tangent to the wave surface. 



To deduce the equation to the surface we may solve (3) to find r, 

 and substitute this value in (4), which will give the equation 



(k - mx - nyf = ^ j (c- + b^) nf + (a= + e) n' + a^ + Ir 



+ ^[(c' _ h') m'- + {a' - e) n" + (rr - ¥)']' - 4 {& - Jr) (a' - c') nfn' \ . 



And if we differentiate this equation first with regard to m and then ii, 

 and eliminate m and n between the three equations, we shall obtain the 

 equation to the wave surface. This is the method which M. Ampere 

 employed with success. 



Instead of eliminating v at first, we may differentiate (4), considering 

 t) as a function of m and n determined by equation (3), we sliall thus 

 obtain the equations 



(a= - v) {c- - V-) ir + (b' - V') {c' - v") m' + (a' - v-) {¥ -v') = ... {\), 

 {% — mx-ny)- = v''{l +»«- + w-) (2), 



