238 Prof. R. Clausius on a new Mechanical Theorem 



The case is still more general where the points do not describe 

 closed paths, but, though the coordinates of the points change 

 in a periodical manner, the periods have various durations^ and 

 the durations may change in different proportions at the transi- 

 tion from the one motion to the other. 



This case can be further enlarged thus- — that periodical changes 

 are not ascribed to the coordinates themselves, but it is merely 

 assumed that the coordinates can be represented as functions of 

 some quantities which undergo periodical changes. 



Finally the treatment can be made still more general, by not 

 directly assuming concerning the quantities by which the coor- 

 dinates are determined that they accomplish their changes pe- 

 riodically, but fixing a less limiting mathematical condition, 

 which is satisfied by periodical changes, but can also be satis- 

 fied without the changes needing to be periodical. This is the 

 method we shall select. 



3. Before proceeding to this treatment of our subject, some 

 mechanical considerations may be premised which will facilitate 

 the understanding of it. 



Given a system of material points whose masses are m v m 2 , 

 &c, which move under the influence of forces possessing an 

 ergal. If the positions of the points are determined by the rect- 

 angular coordinates x x , y lt z v a? q , y q , z 2 , &c, the ergal U is a 

 function of these coordinates. The vis viva T of the system, if 

 we indicate the differential coefficient of a variable, taken ac- 

 cording to the time, by an accent (thus for example putting 



-— - 1 = #/), is expressed as follows : — 



T=S|(^ 2 + ^ 2 + ^ 2 ). ..... (2) 



As is well known, there is a simple relation between T and U. 

 Tn order to be able to write this, the sign to be chosen for the 

 ergal U must be fixed more closely. Usually this sign is taken 

 so that the differential of U represents the work done by the 

 forces with an infinitely little displacement of the points, and 

 hence that the proposition of the equivalence of vis viva and 

 work is expressed by the equation 



T = U + constant. 

 In the form of the proposition, however, which (especially 

 through the beautiful researches of Helmholtz) has more re- 

 cently come into use, and in which we are accustomed to name 

 it the theorem of the conservation of energy, it is more conve- 

 nient to introduce the ergal U with the opposite sign, so that 

 the negative differential of U represents the work, and hence we 

 can put T + U = constant. 



