332 APPENDIX. 



If then Sf 2 4- Srj* + Sf 2 = /> a , a small given quantity, we have, 

 neglecting powers and products of -y- , -y- ... 



/<#y <foA ~ (v _ fdw du\ ^ ^ fdu dv\ 



+ 2 [-r + -T-)$y &s + 2 -J- + -J- Ss&e + 2 -r- + -r 

 V<& dyj * \dx dzj \dy dxj 



or p* = (1 + 2sJ $x* + (1 + 25 2 ) % 2 + (1 + 2s 3 ) 82 2 + 2a % Bz 



+ 2/3 &s&c + 2780%. 



It hence appears that all particles which, after displacement, 

 lie at a given distance from P , must lie before displacement on 

 the surface of a certain ellipsoid of which the centre is at P. 

 Hence, in general, the force called into play by the displacement 

 must be a function of the six coefficients involved in the equa- 

 tion of this surface referred to P as origin. 



But, if the medium be homogeneous, the force thus called 

 into play will be independent of the position of this ellipsoid, 

 and will depend upon its form and magnitude only, that is, will 

 be a function of the lengths of the axes of this ellipsoid. But 

 the reciprocals of the squares on the semi-axes are the values 

 of X given by the equation 



X / > 2 +(l + 2 5l ), 7, /3, 



7, -Xp 2 + (1 + 2*,), a, =0, 



A , _x / > 2 +(l + 2s 



and are therefore expressible in terms of 



4 



Hence, if the force function be called </>, = 1 + < 2 + </> 3 + ... 

 we have 



- (a 2 + /3 2 + 7 2 )}, 

 C 1 +s 2 +* 8 ) 3 + Z> ( 5l +5 2 + 53 ) {4 (, A + Vl +* A ) - (a 2 + /Q 2 + 7 2 )) 

 + E {4 5lVa - ( 8 



