STRAINED ELASTIC SOLIDS 



467 



space bounded by a surface s', then the three following rela- 

 tions are true 



^, dx' dy' dz' = \ a'4> Ds', 

 ox I 



30 

 dy 



-, dx' dy' dz' = / l3'(i> Ds', 



^ dx' dy' dz' = / y'(}> Ds', 

 dz I 



where the volume integrations are to be taken throughout the re- 





(i'K'\) 



Fig. 7 



gion bounded by s' and the surface integrals over s' . Figure 7 

 illustrates the proof of the first equation. The region is divided 

 by up into elementary columns parallel to OX' , whose sections by 

 planes parallel to OY'Z' are elementary rectangles, bounded by 

 sides parallel to OY' and OZ' . Let us integrate {d(f)/dx')dx'dy'dz' 

 throughout that part of the region contained in one of the 

 columns which intersects the surface in two elements of area 

 Dsa and Dsb' at the points A and B; the result is in the limit 



equal to the product of the definite integral / {d(}}/dx')dx' by 



Jb 



the sectional area of the column. Now the definite integral is 



equal to <}>a — <i>B, where (J)a and 0b are the values of 0(x', y', z') 



