768 

 1 d 



2c^ dt 



being zero on account of (14(7) when terms of liiglier order than the 

 second are neglected. 



§ 5. The equation of energj' (together with tiie integral of angular 

 momentum) of the motion of a 'particle in I he field of two fixed 

 centres can be obtained rigorous in the following way. 



It is clear that, whatever may be the influence of both centres 

 on each other, their field will be symmetrically situated, about an 

 axis. Choosing this axis for axis of x and 'calling r the distance of 

 the particle from that axis, if- the angle between a fixed plane 

 through the axis and the plane through the axis and the particle, 

 we obtain 



L =r (ux'^ -(- 2p,vr -\- vr'' -\- qr^t/^ -{- s)i, 



u, p, Vy q, s being functions of .v and ;•. This will yield at once the 

 equations of motion 



d /'dL\_dL d /'dL'\_dL d/dL\_ 

 dt \dw J ~ d.v ' dt\dr J~ dr ' dt \dff J ~ 

 From the third we get 



^ = V^--^/', (17) 



Orf' I J 



and by multiplying them in succession by x, r, ip and adding them 

 we obtain 



^ = /'; (18) 



Here h and A are constants. From (17) and (18) it follows that 



=^'r=-A (19) 



s 



(18) represents the equation of energy, (19) the equation which cor- 

 responds to the integral of angular momentum. With the approxi- 

 mation with which we have contented ourselves in the former §§ 

 g =: — 1 and 



s = c" (1 — 2w + f "'^)' 

 SO that we obtain 



rV/-' =^(;' (]— 2?<; + |«ü'') (19a) 



Without difficulty one sees that (18) agrees with (16(?). (19) enables 

 us to eliminate the variable <f fi'ora the equations of motion. 



