430 RICE 



ART. K 



and one of steel of the same section in different ratios; in both 

 cases the Xx stress constituent is the same, but the en strain 

 coefficient is different (the axis of x being supposed to be 

 directed along the length of the wire). Obviously any complete 

 theory would place at the disposal of the investigator the 

 means of calculating in any given case, the strains which result 

 from the imposition of definite external forces. Equations (29) 

 are differential equations which connect the external forces 

 with the stresses, so that with sufficient knowledge of these 

 forces and of the state of stress at the surface of a body we can 

 in theory determine the stress at any other point of the body. 

 But this will not lead to a knowledge of the strains at each 

 point unless we have a sufficient number of algebraic equations 

 connecting the stress-constituents with the strain-coefficients. 

 So far we have relied on the mathematician to develop the right 

 conceptions and deduce the correct differential equations; we 

 now have to turn to the experimenter who by subjecting each 

 material to suitable tests determines the various "elastic con- 

 stants" of any given substance. This is a matter on which 

 little can be said here, but provided the tests do not strain a 

 body beyond the limits from which it will return to its former 

 condition without any "set" on removing the external forces, it 

 is found, as a matter of experience, that there is approximately 

 a linear relation between strain-coefficients and stress-constit- 

 uents. Under these conditions the deformation of solid media is 

 relatively so small that, although a rectangular element is in 

 general after the strain deformed to an oblique parallelopipcd, 

 the various angles have been sheared from a right angle by 

 relatively small amounts, and we can use the coefficients en, 

 en, . . . 633, referring the system to the same axes before and 

 after the strain. As we have seen above, the pure strains 

 depend actually on six quantities, en, e^, 633, 623 + ^32, esi + en, 

 en + 621, as the rotations are not a matter of importance; 

 furthermore there are only six numerically different values 

 involved in the nine quantities Xx, . . Zz Let us therefore 

 introduce for convenience a small modification of the sym- 

 bolism, and write 



