29] VARIATION AND DIFFERENTIATION. 79 



The terms gp 1; 2 (p 2 , s k (p k are called the first, second and 

 & th variations of cp and are denoted by 



If for cp we put successively x, y, 2, x', y\ z\ . . . we get 



8x =e|, 8y = ey, ds = e$, $*x = d*y =6 2 z =0, 

 8x' = g', <ty' = V, <**' = *', <?V = tfV ==dV =0, 



We thus see that the variations of x, y, 2 are infinitesimal arbitrary*) 

 functions of i, the independent variable, and from the last equation 



that is, the operation of differentiation by the independent variable t 

 and variation are commutative, for the variables #, y, z. 



If we consider qp as a function of the variable 7 the develop- 

 ment by Taylor's theorem for one variable shows that we have for 

 all values of & 



so that 



Now the two variables s and ^ are totally independent of each other, 

 which may be indicated when necessary by writing the derivatives 

 with respect to t as partial derivatives. Now since we may (subject 

 to the usual limitations as to continuity) permute the order of 

 differentiation, we' have 



_a^_aV = _^?> 

 df 8 e * da* a/* 



Multiplying by * after having put s = after differentiation, this 

 becomes 



^U^=<A 



dt dt l 



so that the operations of differentiation by the independent variable, 

 and of variation, are commutative for any function. (It is to be 

 distinctly noted that this holds only for derivatives by the independent 

 variable, that is the one whose variation is assumed to be zero. If 



1) The functions are arbitrary because the functions 6r,, 6r 2 , G s are quite 

 independent of F^, F% , F 3 being taken entirely at pleasure. 



