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IUN7- 

 28, 29] DEFINITE INTEGRALS. 49 



For every value of e we shall have a particular curve for e=0 

 we shall have the original curve, for e = 1 the final curve, and 

 for intervening values of e other curves. A small change in e will 

 cause a small change in the curve, and if e is infinitesimal we shall 

 call the transformation an infinitesimal transformation. The 

 changes in the values of x, y, z, or of any functions thereof, for an 

 infinitesimal change e, are called the variations of the functions, and 

 are denoted by the sign 8. 



Suppose we denote 



dx d*x d k x 



_ _ _ PTif 



dt' dV ' dt k 

 by the letters 



x' y x", <*>, 



and by < any function 



(t, x, y, z, x', y', z', ...... #<*>, y, z^, ...... <>, y, *<>), 



and consider the change in </> made by an infinitesimal transfor- 

 mation, where we replace x, y, z by 



where ^, 77, f are arbitrary continuous functions of t. 



dx ,. , , dx d% ,d^x, d k x 



Then Tt or x is replaced by ^ 4- ej and - by + e 



i.e., by a?*>+f<*>. 



Hence <^> becomes 



<f> (t, x + cf, y + 697, * + ef, ^' + ef, / + /, 



which developed by Taylor's theorem for any number of variables, 

 gives on collecting terms in equal powers of e 



e 2 



where 



W. E. 



