50 DEFINITE INTEGRALS. [INT. III. 



The terms e^, e 2 </> 2 , *$& are called the first, second, kih varia- 

 tions of (j> and denoted by 



If for <f> we put successively x, y, z, x', y' } z ...... we get 



Sy*> = ^* =0. 



We thus see that the variations of x, y, z are infinitesimal 

 arbitrary functions of t, the independent variable, and from the 

 last equation 



that is, the operations of differentiation and variation are com- 

 mutative, for the variables x, y, z. 



It is evident that <& is the &th derivative of < with respect to 

 e for the value e = 0. 



Since we may always change the order of differentiation, it is 

 evident that the commutative property holds for any function. 



Let us now find the variation of the integral 

 ,y,*,',y,/, ...... )dt. 



Changing as to x + 8%, y to y + y, x' to x' + Sat', etc., 

 7 + S/ + iS 2 /+ ...... = f(< + S</> + JS 2 4>+ ...... )dt, 



Jto 



and the variations are 



to 



that is, the operations of variation and integration are commu- 

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



These two principles of commutativity of 8 with d and I form 

 the basis of the subject. 



30. Line and Surface Integrals. If we consider any curve 

 in space joining a point A to a point B, and if on the curve 

 between A and B we place n l points, p ly p 2 ..... .p n -i, whose 



