LAGBANGE'S EQUATIONS 331 



Equation (156), namely 



may be written in the form 



and this, we now see, may be replaced by 



?*^ + ?*'^*- a < 162 > 



Now we have 



sothat 



and, integrated by parts, this becomes 



Since the disturbed configuration, by hypothesis, coincides with 

 the actual configuration, we have 80 l = at times t r and t z . Thus 

 the first term in expression (163) vanishes, and leaves 



, t 



Equation (162) now assumes the form 



The limits t l and t z are entirely at our disposal ; the equation is 

 true whatever values we assign to them. In other words, the sum 

 of a number of small differentials vanishes, no matter how many 



