( 395 ) 



^•" + q^ ,f rrz O, . ■ . . . . . (CJ 



/ + '/, y = 0. . . . . . . . . (Q 



From (his ensues 



y ,c — .?; y =^ y) 

 or 



v' ,/; — ■ ,r' 1/ =: c (14) 



If we take ,/■ and // I o he (rectangular) coordinates then (11) 

 represents a certain curve. To each point of this curve belongs a 

 certain ./■, hence a certain t^. If we now consider the radius vector 

 connecting- a |)oint of the curve with the origin 0, then it will 

 describe an ai'ea whilst describing the curve of wdiicli the element 

 (IS is e(|ual to hi^vdij- ij dv). So for equation (14) we can wi-ite : 



dS 

 2 — = c 



or 



2 



t,^ — S, (15) 



c 



when the constant of integration is put equal to zero in connection 

 with the choice of the direction zero of the radius vector. 



The ecpiation (^15) now expresses that the canonical variable t^ Is 

 proportional to the area descrihed by the radius vector out of 0. 



If we kee[) in mind thai 



we then have according to (14) 



{:'•'! X — <l)''' = <'. • (16) 



whilst from [C^) ensues 



Elimination of ,/" with the aid of (Cj furnishes then 



^fxx 



\\\ eliminating ,/'' out of (16) and (17) w^e tinallx' ai-i-ive at 



C'(fxx 



{'^'Ix—<l) 



Foi- dt^ we lind the expression 



17) 



(18) 



1 



dt^^=-- (xffi — ff) d.v, (19) 



c 



from \vhicli we conclude that ^/ic^ <'////t'/v^////fz/ equation {C) ca?i be solved 

 bi/ one single quadrature as soon as is l- no urn inJdch connection there 

 is betirec)) tiro purtunilar ii/Jegrals. 



27 

 Proceedings Koyai Acad. Ainsterdam. Vol. XIV. 



