130 The Eev. S. Haughton on a Classification of Elastic Media, 



normal and transverse vibrations ; this restriction will reduce the constants 

 from twenty-one to seven ; six of which belong to the transverse vibrations, 

 and the seventh to the normal vibration. It is to be remarked, however, that 

 a function of (\, /i, v, 0, x> 'f ) ^^^^ represent only a part of the properties of 

 bodies whose molecules attract and repel each other. Such a body is always 

 capable of transmitting normal vibrations, and though it may transmit trans- 

 verse vibrations following the laws of Feesnel's wave-surface, yet the normal 

 vibration cannot be supposed to vanish. If we include normal vibrations, the 

 most general form of the subordinate function will be 



V=F{o,,\fx,u,cp,x,ir). (35) 



The manner in which I have obtained equation (35) is not so direct as 

 Mr. Green's method, and I have only used it for the sake of the intermediate 

 equations (31), which exhibit a remarkable property of the quantities (\, /z, v, 

 (p, X, -f )• The direct method of deducing it is the following. Let M. Cauchy's 

 elhpsoid (17) be constructed for the function V, which is homogeneous and of 

 the second order with respect to (a, j3, 7, u, v,w). 



Equations (16) determine the directions of molecular vibration ; in these 

 equations, if {I, m, n) be substituted for (cos a, cos (3, cos 7), we shall obtain 

 the following: 



{Q! - E) mn + F' (n' - m^) + H'ln - G'lm = 0, 

 (R - F) nl + G' {P - n^) -i- F'ml - E'mn = 0, 

 {P - Q) Im + H' {m' - I') -f G'nm - F'ttI = 0. 



These equations express that one of the axes of the ellipsoid is normal to the 

 wave-plane, and consequently that the other two axes are contained in the wave- 

 plane : by stating analytically that these conditions are true, independent of the 

 position of the wave-plane, we obtain the following relations among the coeffi- 

 cients of V, 



(^u) = 0, (av) = 0, {aw) - 0, 



(7m) = 0, (7y) = 0, (/3m;) = 0, 

 {all,) + 2{vw) = 0, {^v) + 2{uiv) = 0, {rw) + 2{uv) = 0, 



{a') = (,3'-) == (r) = 2{u') + {M = H^l + («7) - H^'-) + («/3)- 



These fourteen equations will reduce the function V to the form 



