SURFACE MOTION FOR ANY POSITIONAL FIELD OF FORCE. 169 



/g^x ^n^ { (to+ 7it'^+ lic' ^) + (X«+ X./) {y'- co) H 1 + / 2} 



3(1+ wr') 



We may finally state 



Theorem 9. The most general triply infinite system of curves on a 

 surface possessing Properties I, II, and III is defined by a differential 

 equation of the form 



(56) (a;- v')G'^ 6' (70+ 7^+ 12V'-- 3i'") 



invohing four arbitrary functions 70, 71, 72, co of u, v. 



By comparison of (56) with the differential equation of the tra- 

 jectories 



(11) (lA - cj>v')G'= G{(rPu+ 2K4>) + i^v- 0U+ 2KrP- 2X„c/>)/ - 



(0, + 2X.i/')r"'-3#"}, 



invohing only two arbitrary functions of u, v, we note that Properties 

 I, II, and III are not sufficient to characterize the system of tra- 

 jectories. We may here note the similarity in form of equations (11) 

 and (56). 



§ 7. The Lines of Force. Curves with Property I, II, III, IV. 



On the surface, a line of force is a curve such that its tangent line 

 at any point has the direction of the force vector through that point. 

 The lines of force thus form a simple system of 00^ curves defined by 

 the differential equation 



(57) z/= ^/0. 



Employing (39), we find for the geodesic curvature of the line of force 

 passing through the point 0, 



/ggx 1 ^ 0V«- ^v,.+ 0iA(iA,.- 0j - (x,.0 - K^) (0-+ r-) 



How does this compare with the curvature of the unique hyperoscu- 

 lating trajectory passing through in the direction of the force vector? 

 The value of v" corresponding to a hyperosculating trajectory is given 

 by (43); introducing this and the direction v'= ^/cj) into (39), we find- 

 for the required geodesic curvature, 



/.gx i _ (/)V.- f-0.+ H^jyP,,- (/)„) - (X.0 - K^P) (0^+ ■A^) 

 R ' 'Se\<p-'-\- r)^ 



