30 Prof. A. Gray on Canonical 



ways specified above, we obtain 

 dp . Bp - , , Bp - 



__ _ /BH ^oi <^[ d«2 , , BH B«*\ , . v 



Vd^i B? Ba 2 d</ ' B«a- B<? /' 



which may be written in the form 



b 2 s . ; b 2 s . = b 2 s b% | b 2 s b^ 



B^/iB^ 1 ~d < ] k ~d < i qk d«iB£ B? '" B^BsBg' 



Similarly from the equations 



^£_BH B£_BJI 



rf* == Bp ' B* ~ Bp ' l j 



we obtain 



ys' . a»s' . _ ys' a&j ys' y^ 



Instead of differentiating with respect to £ as in (3) we 

 may do so with respect to any of the constants. In the 

 canonical equations H is — BS/B* 1 in one case and BS'/B^ in 

 the other, and is expressed in each case as a function of 

 {t, p, q). Now consider in the same way — BS/B^-, or 

 BS'/B^, as a function of (t, p, q). The former is 6 t -, the 

 latter is — a v This suggests putting in (3) above h i for H, 

 and taking q lt q 2 , . . . , q k on the left as standing for B^i/d^, 

 B^/Bttj, • • • • , ~d<ljj~d<x>i' so that we obtain instead of (4) 



Bp B^i { Bp dff& = /3k 3% . Bk 3%\ . g . 



B^i B<^ " " ' B^ A . B«i \B% B<? " ' ' B« fe ~dq )' k ' 



Adding to this the equation (2) § 5, namely, 



Bp___3k . 



Ba;~ 3?' W 



we obtain 



©-(I) <■»» 



Similarly by means of the second function S' we deduce 

 the relation 



(!)--<!?> <") 



and the remaining two relations stated in § 6 above can be 

 found in like manner. 



