82 VECTOR ANALYSIS. 



as in the ordinary calculus, but we must not apply these equations to 

 cases in which the values of < are not homologous. 



183. If, however, F is any constant dyadic, the variations of tT will 

 necessarily be homologous with tT, and we may write without other 

 limitation than that F is constant, 



(2) 

 (3) 



dlog{tT}_i 



~sr -f 



A second differentiation gives 



i = P.' r , ' - (5) 



(6) 



(7) 



184. It follows that if we have a differential equation of the form 



the integral equation will be of the form 



p representing the value of p for t = 0. For this gives 



P T <F ~' r 

 dt~ l ' e '?- r ^' 



and the proper value of p for t = 0. 



185. Def. A flux which is a linear function of the position- vector 

 is called a homogeneous-strain-flux from the nature of the strain 

 which it produces. Such a flux may evidently be represented by a 

 dyadic. 



In the equations of the last paragraph, we may suppose p to 

 represent a position- vector, t the time, and F a homogeneous-strain- 

 flux. Then e tT will represent the strain produced by the flux F in 

 the time t. 



In like manner, if A represents a homogeneous strain, (logA}/ 

 will represent a homogeneous-strain-flux which would produce the 

 strain A in the time t. 



