80 IH. GENERAL PRINCIPLES. WORK AND ENERGY. 



some of the variables x, y, 2, . . ., should depend on others, we might 

 require the variation of some of their derivatives, for instance d (~-\ 



We must then, since both functions x and y are varied, introduce 

 the independent, or unvaried variable t, writing 



dy dy dx y' 

 dx = ~dt Tt = x'' 



and performing the operation of variation on the quotient y'/x', 



dx) \x'J x' 



But for derivatives by t we have 



jsv r */dy\ d ~ ^ . ,/dx\ d 



$ y = s uf) = di *9> ** = * U) - <n Sx > 



so that we may write 



A ( d y\ _ d8 y I dx _ d dSx /dx\* 

 \dx) dt I ~dt ~di ~dt~ \di) 



or, once more removing t from explicit appearance, 



* /dy\ _ ddy dy ddx 

 \dx) dx dx dx 



If x is the independent variable, dx = 0, so that we have the same 

 formula as before.) 



Let us now find the variation of the integral 



, x, y, z, x', y 1 , /, . . .) dt. 

 Changing x to x + dx, y to y -f dy, x' to x 1 + dx', etc., 



7+ dl+ (? 2 / + - =< -f d 4- 

 and the variations are 



- =f(<p -f 



*i #1 



dl=fd<pdt, d k l= fd k ydt, 



that is, the operations of variation and integration are commutative. 

 (The limits have been supposed given, that is unvaried.) These two 



principles of commutativity of d with d and I form the basis of 



the subject of the Calculus of Variations. 



(As in the case of derivatives, it may happen that we wish to 

 examine the integral with respect to a variable whose variation does 



