and the Laws of Plane Waves 'propagated through taem. 1 3 1 



2V=.Ao,-'- +P\+Q^ + 11^ + 2i^0 + 26'x + 2i/>f.. (35, a) 



This is the function used by Mr. Green, and is a particular case of equation 

 (35). The first term of this function will determine the normal vibration, and 

 the last six will represent, as we have seen, transverse vibrations propagated 

 according to the laws of Feesnel's wave-surface. 



If the body be homogeneous and uncrystalline, we shall have the relations, 



F^Q, G = 0, H=Q, 

 which will reduce the function V to the form 



2V=Aa?+P{\ + f, + ^), (.gg^ 



This function will represent homogeneous solids, hquids, and gases, and will 

 be the complete function for these bodies, provided there be neither external 

 forces nor pressures at the limits. If there be such forces, however it 

 will be necessary to add to the function (36), which is of the second order 

 other terms of the first order, as in equations (15). If the function (36) be 

 assumed to represent aU the forces engaged, the equations derived from it 

 will represent tlie motion of a body abandoned to its own molecular actions 

 and freed from all external influence, such as gravitation, pressure of an atmos- 

 phere, &c. The known properties of solids, liquids, and eases, enable us to 

 determine the form of the function (36), and thus lead to the terms to be added 

 in the general case for each species of body. 



It is generally admitted that a solid body, if abandoned to itself, would be 

 capable of vibratory motion, and that its molecules, if displaced, would tend to 

 return to their former position. A gas, if abandoned to itself, would be dissi- 

 pated by the repulsive force of its molecules, so that in this case the function 

 (36) should lead to an impossible result, as vibratory motion is impossible 

 without the addition of pressures at the limits, or some equivalent forces. A 

 liquid occupies a position intermediate between a sohd and a gas ; and if we 

 assume that a liquid abandoned to itself will be in a state of unstable equili- 

 brium (i. e. its molecules, if displaced, will not return to their original position 

 while, if undisturbed, they will not be dissipated), we shall obtain from (36) 



s2 



