Relations in General Dynamics. 23 



values a l5 a 2 , ...., a*, &i 5 ^ ••••? fe, say? of the <^s and jo's, 

 H can be expressed for time t in terms of the a's, the &'s, 

 £ and t, or, if we please, in terms of the _p's, the q's, and t. 

 It is desirable for clearness to state explicitly in all dyna- 

 mical relations which are obtained what are the variables in 

 terms of which the different quantities are supposed to be 

 expressed. 



2. It is not unusual to assume tacitly that the reciprocal 



relations discussed in §§ 6 11 below, when established, in 



form, as consequences of the fact that two successive partial 

 differentiations of a certain function of the initial and final 

 coordinates for momenta) are commutative, hold also when 

 the quantities of which derivatives are taken are quite 

 differently expressed. For example from the relation 



&—* < l > 



where S is a function of the ^'s and the a's, and therefore 

 so also is l>i, we get. since dS/dg = p, 



dp = _M< (2) 



Now the dynamical relation which is of real practical 

 importance is one of exactly the same form, which holds 

 whenp is expressed as a function of the initial coordinates 

 and momenta and t, while hi is expressed as a function of 

 the final coordinates and momenta and t. It is one of the 

 objects of the present paper to supply the necessary proof of 

 the permanence of form here exemplified by a particular 

 case of a very general property of canonical relations. 



3. The determination of the complete integral of (2) § 1, 

 consists in finding S as a function of g l9 q 2 , ...., <?*, t, and 

 k coordinates a l3 a 2 , ...., a k (which may be the k initial co- 

 ordinates, or any k independent functions of these) and t, 

 the initial value of the time, which, if it is convenient to do 

 so, may be taken as zero. The finite equations are then 



^- = &, ...., -- -=£,_. (1) 



If a 1? « 2 , ...., a k , the initial coordinates are used, the finite 

 equations become, as is well known, and will be shown in 

 what follows, 



o«i da/,- 



