396 RICE 



ART. K 



notions of molecular structure and also conceive that the 

 material of the body is "smoothed out" to become a continuous 

 medium. We picture a "physically small" element of the 

 body around a particle, i.e., an element of volume small enough 

 to be beyond our powers of handling experimentally and yet 

 large enough to contain a very great number of molecules; the 

 quotient of the mass of the molecules contained within this 

 element by its volume being regarded as the density at the 

 point. 



If a body is strained, obviously some of its particles must be 

 displaced from the position previously occupied in the system of 

 reference. Yet displacement may not produce strain. Clearly 

 there is no strain if each particle receives a displacement equal 

 in magnitude and direction to that to which all the other 

 particles are subject. Again a simple rotation, or a motion 

 compounded of a simple translation and a simple rotation, will 

 produce no strain. In short, strain involves not only displace- 

 ment but also a difference of displacement for neighboring 

 particles (which is not compatible with a simple rotation), and 

 the business of the mathematician is to determine the most 

 convenient mathematical way of stating how this difference of 

 displacement varies for two neighboring particles P and Q 

 supposing that one of them, P, is kept in mind all the time while 

 the other one, Q, is conceived to be in turn any one of the other 

 particles in an element of volume around P. If this statement 

 when formulated turns out to be quantitatively the same for all 

 the elements of volume, we call the strain "homogeneous;" 

 otherwise it is "heterogeneous." 



We will consider (with Gibbs) that the body is first in a "com- 

 pletely determined state of strain," which we shall call the 

 ^' state of reference." Let P' be the position of a point or particle 

 of the body in this state. It is then strained from this state, 

 and we denote by P the position of the same particle. Consider 

 another particle, near to the former, whose position in the state 

 of reference is Q' and after the strain is Q. The mathematical 

 formulation of the nature of this strain will summarize all the 

 essential information concerning the elongation of the element 

 of length P'Q' and also its change of orientation when it is dis- 



