J8i*] 



resulting from Hamilton's Theory of Motion. 285 



it makes 



and the second and third terms in equation (6) can each, in rela- 

 tion to the coordinates, be conceived as an explicit alteration 

 according to t which may be expressed by 



? 



Then comes 



d{v+w) r 5(V+W) -i_ 

 dt L s* J ' • " ' {) 



and this is the proposition sought : The sum of the alterations 

 in, the primary function and the force-expenditure } which are pro- 

 duced by the variation of the initial and final coordinates alone, is, 

 in the valuation of every motion that presupposes a force-function 

 and neither contains the time explicitly in this nor in the conditions, 

 equal to nil. 



As the variation of the motion is only subject to the condition 

 that it does not destroy the limitations of the system, but in the 

 rest, as already shown, may very well take place under accession 

 of energy, the proposition we have gained is independent of the 

 special kind of the accession. In this relation the coordination 

 with the proposition of the vis viva, which likewise gives the in- 

 crease of the latter independent of the kind of variation of the con- 

 figuration, is evident. But this independence forms only one side 

 in the latter proposition ; it has received, as is known, another, 

 more important, through the remark that the force-function (in 

 the above representation) is nothing else but the potential energy 

 of the system. In such a new direction the new proposition 

 shall now be investigated. 



The sought signification of the primary function readily ap- 

 pears if we give up the forces on which the ordinary theory of 

 motion rests, and introduce in their place momentary impulses 

 capable of producing the velocity existing at any instant. That 

 such a manner of consideration stands in essential connexion 

 with Hamilton's theory of motion has not yet, so far as I know, 

 been rendered evident, although Thomson and Tait have recently* 

 drawn attention to the importance of this second method of pro- 

 cedure, not inferior to the first, and have more nearly completed 

 its theory. In it the components of the momentary impulse 

 (formed according to the general coordinates), if the components 

 of the forces taken according to the rectangular coordinates be 



* Treatise on Natural Philosophy, pp. 206 et seqq, 



