Relations in General Dynamics. 31 



10. The passage from d?/d*=dH/dj9, dp/d*= -BH/cty, 

 (with £>=d8/dg) where H is a function of the ^'s, t, and 

 ^ constants, to dq/dt^'dH/'dj^^ dp/dt= — ^H/^^, where H is 

 a function of (j;, 9-, £), and the establishment of the four 

 relations given above, were effected by Jacobi ( Vorlesungen^ 

 p. 395) by a direct process which seems to have escaped the 

 notice of most of those who have written on the canonical 

 equations. With some modifications it is shortly as follows. 

 The substitution of the values of the a's in "d$/"dq in terms 

 of (p, q, t) must lead to the identity p=p. Hence differen- 

 tiating on the supposition that this substitution has been 

 effected, we get 



~dq §«i d? ~da k ~dq ' • • K J 



There are h such equations. Multiplying them in order 

 by tyifo a i> "dq^dap , and adding, we get 



~dq cK- ~dq "d-dt ' "bq cK- »■ * w 



where R is the sum of all the terms except those which are 

 the first terms on the left of the equations. 

 But we have also 



H--* (3) 



and it is clear that if we substitute in dS/Ba t - the values of 

 the coordinates in terms of the 2k constants and t, this 

 equation will reduce to the identity b i — bi. We have 

 therefore k equations of the form 



KdaJ 



MjY a*s v ys. a gl| } a 2 s *g, =0 



~daj "ba^a. 'dq i 'da j 'da i bq k ba j 'da i ' { ] 



Multiplying these in order by B«i/d</, "daz/bq, . . . . , (where 

 the a's are regarded as functions of the variables (p, q, £)), 

 and adding the set of equations together, we obtain 



B 2 S -da, t B 2 S B« 2 , , ~d 2 $ ba k 



oai^a^q da 2 oa i dq O^oa^q 



where R has the same value as in (2). 



Writing now bp/bqi, bp/bq?, • • • > instead of "bp^fbq, 

 'dp2J'dq, . . . . , in (2), adding on the left of that equation 

 9p/3a i5 and on the left of (5) the equal quantity VS/oq'da i% 

 we get 



©»-© « 



