162, 163, 164] ENERGY OF FIELD. 427 



If, as in 131, we draw surfaces close to the surface distribu- 

 tions, and exclude the space between them, we may, as above, extend 

 the integrals to all other space, so that 



the normals being from the surfaces S toward the space r. But by 

 Green's theorem, as before, this is equal to the integral 



> "- ///!'+ '+>< 



00 



Thus the energy is expressed in terms of the strength of the field 



at all points in space. This integral is of fundamental importance 

 in the modern theory of electricity and magnetism. 



It is at once seen that this always has the sign of y, so that it is 

 positive for electrical or magnetic, negative for gravitational dis- 

 tributions. 



CHAPTER IX. 



DYNAMICS OF DEFORMABLE BODIES. 



164. Kinematics. Homogeneous Strain. We have now 

 to consider the kinematics of a body that is not rigid, that is, one 

 whose various points are capable of displacements relatively to each 

 other. In the general displacement of such a body every point x, y, z 

 moves to a new position x\ y', z 1 , so that x f , y', 0' are uniform func- 

 tions of x, y, 0. The functions must also be continuous, that is, two 

 points infinitely near together remain infinitely near together, unless 

 ruptures occur in the body. 



The assemblage of relative displacements of all the points is 

 called a strain. The simplest sort of strain is given when the func- 

 tions are linear, that is, 



x 1 = a^x -f a 2 y -j- a^z, 



1) y' = \x + \y + M, 



0' = W + %y -f c 3 z, 



where the a's, Vs and c's are nine constants. 



