resulting from Hamilton's Theory of Motion. 275 



strated, is sufficient for its definition. The primary function is 

 a complete solution of this differential equation ; and any com- 

 plete solution of the latter, analogously differentiated according 

 to the constants, gives the system of the integral equations. 

 Henee, in the Hamilton-Jacobi method, the entire problem is 

 concentrated into the one integration of the partial differential 

 equation, in contrast to Lagrange's way of proceeding, in which 

 only single integrals are found by aid of the known principles. 

 The integration of the partial differential equation was developed 

 by Jacobi* generally both in the way already pursued by La- 

 grange and Pfaff, and also by a new and grand method, both of 

 which methods have been adopted in a series of more recent 

 works. 



The theory above mentioned has recently undergone expan- 

 sion in two respects. If the investigation by Hamilton and 

 Jacobi referred to actual space, for which the element of a line 

 proceeding from a point is capable of being represented by the 

 square root of the sum of the squares of differentials of the ordi- 

 nates of the point, Lipschitzf formed a more general conception 

 of the problem, inasmuch as he assumed the line-element to be 

 equal to the pth root of any real positive form, of the pth degree, 

 of the differentials of any coordinates of the point in question. 

 The element of its integral corresponding to the primary func- 

 tion becomes the sum of any form of the pth degree of the dif- 

 ferential quotients, taken according to time, of the variables and 

 any force-function depending only on the variables — this sum 

 multiplied by the time-element ; so that the problem of mecha- 

 nics is changed into a perfectly general one of the calculus of 

 variations. If, further, Hamilton assumed a force-function 

 which depended only on the coordinates of the moved point, and 

 if Jacobi extended the investigation to a force-function explicitly 

 containing the time, Schering J conceived the problem in this 

 direction more generally, introducing forces dependent not only 

 on the position but also on the state of motion of the masses. 

 This dependence is so chosen that, understanding by II the re- 

 sulting force, and by dr the virtual displacement of the mass- 

 points, "ZJidr becomes the difference between a total variation 

 and a total derived according to time ; and this generalization is 

 at the same time accomplished from Lipschitz's enlarged point 

 of view. In it, therefore, motions can be treated which, for in- 



* " Vorlesungen iiber Dynamik : Nova methodus" &e„ Bovcliardt's 

 Journal, 60. 



t " Untersuchung ernes Problems derVamtionsveelinnng," Bovcliardt's 

 Journal, 74. 



X Hamilton- Jacobi'sche Theorie ftir Kv'afte, eleven Maass von derBewe- 

 gimgder Kovper ablmngt/' Abhandl. der Gotting.Ges. derWiss(tosch*\&73, 



T2 



