SURFACE MOTION FOR ANY POSITIONAL FIELD OF FORCE. 171 



§8. Curves with Properties I, II, III, IV, V.— Complete 



Characterization. 



The analytical expression for the final property is most readily 

 found by comparing the coefficients in equations (56) and (11). If 

 (56) is to reduce to (11), we must evidently have 



(63) CO = -; 7o = h iX,.; 7i = h -Xr 2X„; 







Substituting 



i/' = a;0; i/',, = co„0 + a30„; >/'„ = co„0 + w0t., 

 we get from the second and fourth of equations (63), 



(64) ^ = ^ " ~ -^" ~ "^" = (log 0)„; ^' = - 72 - 2X,co = (log 0).. 



CO 



Substituting these values in the third equation (63), we have 



(65) 7o + 7i<^ + 721^" = i^u + coco,) + 2(X, — oAu) (1 + co^). 

 Equations (64) may be combined into 



(66) (72 + 2X„co)„ + h ~ -^^' ~ "" ) =0. 



If, then, (56) is to reduce to (11), the functions 70, 71, 72, co must 

 necessarily satisfy the relations (65) and (66). Conversely, if 70, 71, 

 72, CO satisfy (65) and (66), it is possible, by virtue of (66), to find a 

 function log (and hence 0) to satisfy both equations (64) ; and if we 

 then choose \p = co0, we shall have found a pair of functions 0, \p, or a 

 field of force, which satisfies all the equations (63). We may thus 

 state 



Theorem 12. In order that an equation of the form (56) should 

 represent a system of trajectories xinder some field of force, it is neccssanj 

 and sufficient that the four arbitrary functions of u, v satisfy equations 

 (65) and (66). 



Now (65) is the same condition as (62), and we have already inter- 

 preted this geometrically by Property IV. It remains therefore 

 to interpret condition (66) geometrically and thus complete the 

 characterization. 



