242 Prof. R. Clausius on a new Mechanical Theorem 



tion of which may be defined thus : Bq v is the difference between 

 the value which q v has with the original motion at a certain in- 

 stant and the corresponding value of q v with the deviating motion. 

 But now it is doubtful which of the infinity of values successively 

 taken by q v with the deviating motion is to be regarded as the 

 corresponding value. Hamilton has certainly said nothing defi- 

 nite on this point ; but, by a closer consideration of his deve- 

 lopments and equations, one can easily perceive how the varia- 

 tions therein occurring are to be understood. If we commence 

 with the values which the quantities q lt q 2 , . . . q n with the 

 original motion have at a certain time t, the corresponding values 

 with the changed motion are those which the quantities have at 

 a time t + Bt — in which the variation Bt is not determined, but 

 the same for all quantities. 



That a common value is indeed to be attributed to the varia- 

 tion Bt throughout the system is at once evident from this, that 

 in equation (La) Bt appears as a quantity valid for the entire 

 system. 



Another circumstance which leaves no doubt about this is the 

 following. Hamilton presupposes, in the derivation of his equa- 

 tions, the proposition of the conservation of energy, according to 

 which the sum T + U is constant. But that proposition of 

 course holds only when, in the formation of the quantities T and 

 U, the variables which determine the positions and velocities of 

 the points are brought into the calculation with the values they 

 have at a common time, whether this time be t or t-\- Bt ; but we 

 must not combine values which refer to different times in order 

 to form from them the quantities T and U. Accordingly, 

 with equations thus produced, so long as the opposite is not 

 expressly stated and shown to be admissible, it must be taken 

 as evident that only simultaneously occurring values of all the 

 variables are ever brought into the calculation. 



In order to see how variations which correspond to a common 

 time-variation Bt behave, we will now select a simple case for 

 consideration. We will, namely, presuppose that with the ori- 

 ginal motion all the points describe closed paths, and with the 

 deviating motion, again, all the points, starting from infinitely 

 little-changed initial positions, describe infinitely near-lying 

 closed paths, but that the periods of the different points are 

 altered in different proportions. 



As the time-variation Bt can be taken at pleasure, we will 

 first suppose Bt = 0; that is, we will regard as corresponding to 

 one another such values of the variables as belong to one and 

 the same time. If, then, a point has different periods with the 

 two motions, the two positions which belong to one and the same 

 time, reckoned from the commencement of the motion, are so 



