118 The Rev. S. Haughton on a Classification of Elastic Media, 



SECTION II. — LAWS OF NOEMAL VIBEATIONS. 



It has been shown that the differential coefficients of the function F, taken 

 with respect to (oj, 02, &c.), denote the normal and tangential actions of the 

 surrounding body on an elementary parallelepiped. I shall now suppose that 

 the body is so constituted that the tangential forces vanish, and so deduce the 

 laws of a body capable of transmitting pressure in a normal direction only. 

 The condition that the tangential forces vanish will give the equations, 



^1^0, '^^O, ^^^=0, 

 da, ' «//3, ' dy, 



djL^O, ^^=0, ^^=0 

 da^ ' dp., ' dji ' 



which will reduce the function to the form 



2 T" = .1 ac + Bp? + Cyi + 2P/3j73 + 2Qar(, + 2Rafi, . 



At first sight, it might appear that we might assume this or any other form 

 of fimctiou to define a body, and proceed to deduce the laws of motion ; but, 

 as it is possible that an assumed form of the function may be only true for par- 

 ticular axes of co-ordinates, it is always necessary to examine whether a trans- 

 formation of co-ordinates will introduce any new terms ; if this be the case, 

 tlien the assumed function will not represent completely the molecular structure 

 of the body, as it will be merely a simplification of a more complex function, 

 produced by assuming particular axes of co-ordinates, which are connected 

 ■with the crystalline structure of the body. I shall examine the function just 

 given for normal pressures by this method, and determine it so as to be inde- 

 pendent of the axes of co-ordinates. The relations between the two systems 



of variables are 



,r = aw' + by' + cz", 



y = a'x -t- h'y' + c'z', 

 z = a"x'+b"y'+c"z'; 



also (I, rj, <) will satisfy these equations. Let us now assume 

 M = ^^-t-7,, I' =7,+ a,, W = a., + p^, 



