1208 



becomes the straight tangent, {a is the radius of the sphere of 

 attraction; outside a there is no attraction any more). 



If now the attractive force is in the path = — ƒ (r) (hence always 

 directed towards the centre 0), then the equation 





dt 



d*y d f dx dy\ dx 



j; — ^ = 0, or — I /v "T X -i- \^0, or y — 



dt"" 



dtV dt 



dt 



dt 



dy 

 X — = const. 

 dt 



d*x d*y 



follows from u — = — f(r) cos \b and u — ^ z= — f(r) sin U' with 

 dt* ^ dt* -^ ^ 



x = rcosxp, y = rsinyp (r is the radius vector, n? the amplitude). 



u n <^^ dr , dtp dy dr dtp 



out trom — = cos \b — — r sm rp —r , — := sm U? h r cos xp — 



dt ^ dt ^ dt dt dt ^ dt 



tollows immediately y x~ r= — ;•' — (which is also immediately 



' ^ dt dt dt 



seen), hence 



dxiy 

 dt 



(1) 



in which c is a constant (which is still to be determined more closely). 

 This is the well-known law of sectors. 



Ihe square ot velocity u*=i\-~\ + I t^ I . expressed in pole 

 coordinates, evidently becomes i= ( — J 4- r' | — ƒ . But also we have: 



dy' 



d*x dx d'y dy f (r) 



dt' dt dt* dt a 



dx 



cosxp 1- sin \p 



dt dt 



i. e. because the expression between [ ] in the second member, in 



civ 

 virtue of r* = x* 4- y*, is also = - ; 



dt 



dt 



hence 



:(ï)'HJ)j 



/ {r) dr 

 ft dt 



d (ir) = r dr ^ ;— dr, 



fi IX dr 



when Pr represents the function of force, and ft the mass of a 

 molecule. Integrated between the limits A and P, this gives therefore 

 (in A Pr ifi = 0, and u = w.) : 



ii^{u*-u,*) = 0-P,., 

 or 



P, 



u* = u,* -2—, 



(2) 



