SURFACE MOTION FOR ANY POSITIONAL FIELD OF FORCE. 159 



or if, for convenience, we write 



(6) 



equations (5) become 



X = - log At, and yu = e^^ 



(7) 



u 



-^ — {\uU~ + 2\yui — \uV^) 



e- 



To get the differential equations of the trajectories we must elimi- 

 nate the time from equations (7). We evidently have 



V = 



V" = 



uv — vu 



u 



W' 



and hence 

 (8) 



,.// 



e^H^ 



-(^ - 4>v') + {K -Kv') (I -\- v'^). 



Using the abbreviations 



fir 



(9) 6" = ^ ^ /" + v" (X„ - 2X„ v' + 3X. / 2) 



du 



— (Kv + X„ V' — \uu V' — \uv V' ^) (1 + / ^), 



where 6' = is the differential equation of the geodesies on the surface, 

 we may write (8) in the form 



(10) n^ = tz^. 



Differentiating (10) with respect to u and using (7), we get 



(11) (lA - 0t/) 6" = G{{yPu + 2X,0) + (^„ - 0„ + 2\,rp - 2K<l>y 



- (0„ + 2X„V)/'-3</>^"}, 



as the differential equation of the trajectories. If we replace G and 

 G' by their values from (9), we get a differential equation of the form 



(12) v'" = P -\- Qv" -\- Rv"'- 

 where 



' P = (ao+ axv'-\- a2/2 + a^v" + a^v' ^ + a,v'^)/{rP - #'), 



(13) ] Q = (^0+ /3i/+ ^2V' ' + /33i'' «)/(!/' - <Pv'), 

 R= -2,4>/{yp - (t>v'), 



