SUKFACE MOTION FOR ANY POSITIONAL FIELD OF FORCE. 159 



or if, for convenience, we write 



(6) 



equations (5) become 



1 



X = - log/i, and jx = r 



,2X 



(7) 



u — 



^2X 



{\uU- + 2\vUV — \uV^) 





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

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



, _ v f, uv — vu 



V = —, V = 

 u 



i? 



and hence 

 (8) 



1 



e'~\l'' 



(,A-0/) + (X„-X„ «')(!+«'-). 



Using the abbreviations 



■ G^v" - (X,-X„ /)(!+/ 2)^ 



(9) 



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

 du 



— (Kv + X„„ / — X„„ v' — Kv v'^) (1 + v'^), 



where G* = is the differential equation of the geodesies on the surface, 

 we may write (8) in the form 



(10) -- - "^ - '^^' 



u 



^G 



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



(11) (xp - <t>v') G' = G\i,^Pu + 2X,(/>) + {^p, - <A« + 2X„V - 2X,0)/ 



- (0. + 2X„V)/=^-30/'}, 



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+ ay+ ao/- + aj/^ + a^v' ^ + ay^)/(^P - <l>v'), 



(13) ^ = 08o+ /3y + ^2V' ' + ^sv' ')/{4' - <t>v'), 



. R= - 30/ (i/' - #'), 



