78 IH. GENERAL PRINCIPLES. WORK AND ENERGY. 



transformation of the curve. The change may be made gradually, 

 for example, 



For every value of s we shall have a particular curve - - for 

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

 and for intervening values of other curves. A small change in s 

 will cause a small change in the curve, and if s is infinitesimal we 

 shall call the transformation an infinitesimal transformation. The 

 changes in the values of x, y, 8, or of any functions thereof, for an 

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

 are denoted by the sign d. 



Suppose we denote derivatives by the independent variable , 

 dx d*x d k x 



etc. 



dt dt* dt k 



by the letters 



J /y. " /yi " /vi (A) 



*"'.> * ) "*' J 



and by (p any function of the independent variable, of the dependent 

 variables, and of their derivatives up to the m th order 



and consider the change in y made by an infinitesimal transformation, 

 where we replace x 9 y, 8 by 



y- 



where %,vj, are arbitrary continuous functions of t. 



dx dx , d , d w x , d k x , d* 



Then or x is replaced by --- 1- and r by r + a v 



d* d# d# d* dt k dt k 



i. e., by 



x (k) + ggw. 



Hence p becomes 



ey ey, s fi, * , y , 

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

 gives on collecting terms according to powers of s 



2 A 



(p (t, x, y,g, x\ . . .) + stp + ^(p. 2 - - + fr<p k + , 

 where 



