40 Prof. A. Gray on Canonical 



As an example we take the simple case of v' = q, q'=—2^- 

 AVe obtain 



tp'dq' - tpdq = - 2 [qdp +pdq) + dt(pq), . (10) 



and therefore S = S — ^(pq) is the new principal function. 

 This it will be seen is the second Hamiltonian function for 

 the variables p, q, with its sign changed. 



20. The 2k relations referred to above enable the 2k quan- 

 tities (p, q') to be found in terms of the 2k quantities (p, q), 

 or the 2k momenta (j^'tp) * n terms of the 2k coordinates 

 (q, q). This presupposes that there is no functional relation 

 connecting the 2k quantities ((/', q). Now a relation or 

 relations of this kind may exist, that is, it may be possible to 

 eliminate the p' 's and the p's, from the 2k equations which 

 define the p 's and the q 's in terms of the p's and q's, so as 

 to give equations of the form 



<Kqi, 92, ..., &', gii q 2 , ..., q k )=Q- • • • (i> 



If there are h such relations, then along with 



tp<dg'-$pdq = dW=2^dq' + 2^dg, . (2) 



where W is a function of the variables (q' } q) we have 



1 



[ (3) 



*^+*^*-°l 



2 ^- -, dq + Z -^- dq = 



dy ^q * J 



Hence if Ai, A 2 , ..., \ h be undetermined multipliers we have 



There are 2k equations of this form for (p l , p) and the 

 li relations <f>i = 0, . . . ., <j)h = 0, so that the 2k+h quantities, 

 that is the momenta and the h multipliers, can be deter- 

 mined. 



By the values of p/ } pj we have of course simply 



$p'dq'-2pdq=dW (5) 



