38 Prof. A. Gray on Canonical 



coordinates were independent we should have equations of 

 the type 



§£=* ^ = -'\ ••••(*) 



But now we have at time £ equations of the type 



-n^ =J?i + ^l a il + ^2«J2 +.... + Xm%, • • (3) 



^— = — hi + /Ai&l + /* 2 &2 + + ^ m ft HZ , . (4) 



Oai 



where X 1? \ 2 , . . . ., /a 1? /-t 2 , . . . ., are undetermined multipliers. 

 Thus for a variation from the actual to an adjacent succession 

 of positions without change of £ or r, we obtain 



8S=2(|| Sy) +2 (gs«) =2( i %)-2(&&»), . (5) 



as in the case of a holonomous system. The only difference 

 is that in this case, while the variations from the actual path 

 are all possible, the succession of positions given by these 

 variations is not in general a possible path. 



For the more general variation, as in § 15 above, the 

 result is also the same as that for a holonomous system, with 

 the difference just stated. 



19. The question of permanence of canonical form, for 

 changes of the (p, q) variables, has been discussed by various 

 writers, from the point of view of the theory of contact 

 transformations originated and worked out by Lie *. The 

 use of Hamilton's function adds clearness to this mode of 

 considering the subject. 



If the variables of the type p, q be changed to others of 

 the type p\ q' which are functions of p, q but not of t, and 

 the condition 



2p'dq'-2pdq=d\Y, (1) 



that dW be an exact differential when expressed in terms of 

 the p's and the q's, be fulfilled, the change is called a contact 

 transformation. We suppose for the present that there are 

 2k independent equations connecting the variables p', q' with 

 the variables p, </, and that W has by means of these been 



* See an excellent account by Lovett in the ' Quarterly Journal of 

 Mathematics/ of Lie's remarkable dynamical paper in the Archiv for 

 Mathematik og Naturvidenshab, vol. ii. ; Christiania, Jan. 1877. 



