Relations in General Dynamics. 29 



The device of regarding the time as a coordinate is useful 

 in many respects as we shall see later. It has been employed 

 by Poincare, Minkowski, and others in various applications. 



8. The relations (4), (6), (7) of § 6 appear to have been 

 first published by Donkin (Phil. Trans. 1855), but they are 

 contained iii one of five undated papers by Jacobi (Nach- 

 gelassene Abhandlungen) appended by Clebsch to his edition 

 of the Vorlesungen fiber Dynamik, published at Berlin in 

 1866. Jacobi died in 1851. All possible relations of this 

 kind, including the usual canonical equations, are involved 

 in an extended form of a variational theorem due to Lagrange, 

 which is most easily and directly established by means of 

 the function S, and constitutes a fundamental relation 

 between initial and final coordinates. As stated in the 

 former paper, equations (1), (6) and (7) of § 6 were, in 

 part at least, rediscovered and interpreted by v. Helmholtz 

 (Crelle, 1866), and illustrations were given by Lamb (Proc. 

 Lond. Math. Soc. 1866), who also derived v. Helmholtz's 

 results from Laoran^e's theorem. 



9. Assuming the canonical equations we can now derive 

 two theorems by means of which (4), (6) and (7) of § 5 can 

 be inferred. The proof by inference (the idea of which 

 becomes clearer when t is regarded as a coordinate) may be 

 justified in various ways, e. g. by a process due to Jacobi 

 (Nachg. Abhandl., Vorlesungen, p. 395). In the canonical 

 equations H, as has been stated, is a function of (t, q, p) *, 

 and the constants do not appear. The transformation of H 

 from the form in which it is used in (4) of § 4 is by substi- 

 tution from the equations 



a t : = 5ifei>---* &">>i, -..>/>*,*)> • . . (1) 

 or from 



^%ir-v?^iv,iv0; • • • (2) 



[according as the variables transformed are (t, q, a) or 

 (t, q, &)], which give the initial coordinates and momenta in 

 terms of the final values of these quantities and the time. 

 Now from the equations 



dp_ _BH -dp_ _3H 



dt ~ ~dq ' -dt ~ B</ ' • • ' ' [D) 



where of course H is supposed expressed in the two different 



* Here, and in what follows, (t, q, p) signifies the variables t. q l} 

 q.,, . . . (],., 2hi Pi, ' • • Pfc Similar abbreviations are used in other eases. 



