22 Prof. A. Gray on Canonical 



7. Apart from these speculations, the definite result seems 

 to emerge, that the departure from the classical mechanics 

 is to be looked for in the fundamental equations of tether and 

 electricity. Nothing so complicated as the structure of 

 matter appears to be involved. It is not a question of 

 modifying our ideas (in so far as we have ideas) on the 

 build of atoms or molecules : we are called on to revolu- 

 tionize views which have long been regarded as well- 

 established on the nature or meaning of electricity, a?ther ,or 

 radiation. 



III. On Canonical Relations in General Dynamics. 

 By Professor A. Gray, F.R.S.* 



1. TN a paper on General Dynamics (Proc. R. 8. E., 

 JL Feb. 19, 1912) I have derived Hamilton's principal 

 function S, and the parallel function S' [§ 5, (1) below], 

 with the corresponding partial differential equations, by a 

 direct process not involving the calculus of variations. 

 Hamilton's Principle and the Principle of Least Action may, 

 as exemplified below, be deduced from the functions S, S', 

 and it will be seen that the use of the two functions faci- 

 litates the proof and discussion of various other theorems. 



It may be recalled that the canonical equations of a 

 system, unacted on by f riction, and defined by k coordinates, 

 qi, q 2 , ...., gk, that is the 2k equations of the type 



dp _ B H dq _ dH . . . 



dt~~ jyf dt ~~!yp * 



can, as Jacobi proved, be replaced by finite equations, if the 

 complete integral of the Hamiltonian differential equation 



I +H (1'I— 1| *.*■-. ft.')" • <-«> 



can be found. 

 Here 



E = 2(pq)~T + V, (3) 



and p( = 'dTj'dq) is derived from T, the kinetic energy ex- 

 pressed as a function of the ^s, <^ s ? an d, it may be, of t- 

 Since the solution of a dynamical problem consists in the 

 expression of the values of the coordinates, the velocities 

 (the q&) [or the momenta (the jq's)] in terms of t and initial 



* Communicated bv the Author. 



