62, 63] HAMILTON'S PRINCIPLE. 121 



7\T 



The derivative =, , which i 

 oq s 



the q"s, is generally denoted by 



The derivative =. , which is a homogeneous linear function of 

 oq s 



In the case of rectangular coordinates, 



the ^-component of the momentum of one of the particles. In 

 general, p g may be called the generalized component of momentum, 

 belonging to the coordinate q s and velocity q s '. The equations of 

 motion may be written 



dp, 3T D dT 



= P 



or if, as we shall in future do, we denote by P s simply that part 

 of the impressed force which is not derived from the potential 

 energy, under which are included all non- conservative forces, 



> 



63. Theorem on Reciprocal Functions. The ordinary 

 notation for partial derivatives of functions of several variables 

 sometimes gives rise to a certain confusion, from the lack of 

 indication of what variables are to be considered as constant 

 during the differentiation. For instance, suppose we have a 

 function F of any number of variables, which for convenience 

 we will divide into two classes, denoting them by the letters 



Suppose now we have n functions of these variables, given by the 

 equations 



