resulting from Hamilton's Theory of Motion, 281 



which, in consequence of the supposition made, are not connected 

 with one another by any relation. Hence all the variations 

 Bq ( , 5g?, 8c k become mutually independent. 



In consequence of this, equation (1) can be immediately split 

 up into single equations. Putting, that is to say, the expression 

 which stands under the integral-symbol 



»» -*£*]** 



we get the differential equations of the motion 



» = B(T-U), 



dt ~dq. 



and as, conversely, the latter are demonstrated by Lagrange to 

 be independent of equation (1), it follows that the expression 

 standing on the right-hand side under the integral-symbol va- 

 nishes under all circumstances. Therefore neglecting it, we 

 have 



-sv^fy-s^-J's^M'- 



(2) 



and this is Hamilton's symbolic equation expanded. Because, 

 namely, the variations are all independent one of another, they 

 furnish at once the integral equations 



_dV_ BV_ 



Equation (2), with only an unimportant difference in the way of 

 writing it, has already been given by Clausius*; it is, however, 

 to be remarked that his deduction refers only to motions of which 

 the force-functions and equations of condition do not explicitly 

 contain the time. The form in which it gives the variation 8V 

 is not sufficiently general for the following considerations, because 

 in general the time t likewise varies, and therewith a partial 

 change is produced both in T and in U and consequently also in 

 V, which is neglected in equation (2). 



It shall therefore now be assumed that the time t is no longer 

 the independent variable, but undergoes the change St on the 

 variation of the motion. In order to understand the sense of 

 this variation, it must be considered that the time is not to be 

 variated wherever it occurs, but only where it occurs explicitly ; 

 for a variation of the other would amount to a variation of the 

 initial and final coordinates ; and this is already done. In this 

 case, therefore, the primary function V is taken as dependent on 

 the initial and final coordinates, this explicit time /, and the 

 * Pogg. Ann. vol, cl. p. 122. 



