relative to Stationary Motions. 241 



The second form of the equation is still more convenient in the 

 latter respect. In it the integral 



U (T-V)dt 



I 



is to be looked upon as a function of the quantities q x , q 2 , . . . q n) 

 k v k 2 , . . . k n) and t ; and when this function is known, we obtain 

 through the analysis of the equation alone the first and second 

 integrals of the differential equations of motion. 



5. From the preceding it is evident that Hamilton's principle 

 is one of extraordinary importance for mechanics. Neverthe- 

 less it is, for two reasons, not suitable for our purpose. 



In the first place, great as is its generality in other respects, 

 ill one direction it is not sufficiently general. In the equation 

 there arc two motions compared which differ infinitely little from 

 each other ; and their difference may be reduced to this — that the 

 initial coordinates and velocity-components of the movable points 

 had somewhat different values with the one motion and with the 

 other; but the ergal U is presupposed to be, with both motions, 

 one and the same function of the space-coordinates. The dif- 

 ference, however, between two motions can also be occasioned 

 by the ergal having undergone a change which is independent 

 of' the alteration of the coordinates. In the science of heat this 

 case is quite common, because with a body upon which certain 

 external forces act, under the influence of which its molecules 

 perform their motions, these forces may undergo a change which 

 is expressed mathematically by a variation of the ergal, whereby 

 of course a changed molecular motion is necessitated. Transi- 

 tions of this sort from one motion to another cannot be treated 

 by means of Hamilton's equation. 



The second of the reasons above alluded to refers specially to 

 stationary motions. If a stationary motion as such is to be 

 more closely determined, the question is, not to give the posi- 

 tions and velocities of all the individual points for single mo- 

 ments of time, but much rather to fix the general character of 

 the motion independent of time. An equation which is to serve 

 for this purpose may certainly contain variable terms ; but their 

 variability must be confined to certain fluctuations of their values, 

 which are repeated in a similar manner, so that the equation has 

 to a later time essentially the same relation as to an earlier one. 

 If, on the contrary, terms occur which are undergoing continu- 

 ally greater variations, so that the equation has not the same 

 relation to a later as it had to a former time, this circumstance 

 makes it unsuitable for our purpose. 



We will now consider Hamilton's equation from this point of 

 view. In it occur the variations hq x , 8q 2 , . . . hq* l} the significa- 



PJiil. Mag. S. 4. Vol, 46. No. 305. Sept. 1873. * S 



