166, 167] 



COMPOSITION OF SMALL STRAINS. 



437 



Suppose two small strains take place successively according to 

 the equations 39) for the first, and 



X' 



43) 



for the second. 



Substituting the values of x\ y' } # r from 39) in 43) we obtain 



*4(l+< 



-f 



-f 



Neglecting terms of the second order we obtain the equations of the 

 resultant strain 



-f- -f 



44) ?" = & 



and for the resultant shifts 



45) 



(a 2 -f 



that is, successive small strains are compounded by adding their 

 shifts. This important proposition enables us conveniently to resolve 

 small strains into types already studied. Every small strain represented 

 by equations 40) can be written by addition and subtraction of 

 equal terms 



u = a^x + (a a -f & x )y + (a 3 + cJ0 



46) 



-f 



