SURFACE MOTION FOR ANY POSITIONAL FIELD OF FORCE. 



181 



of the hj'perosculating curves of Property III which have the directions 

 of the u and v parameter curves, (95) becomes 



(96) 



cU 



1 



n + r 



Introducing the abbreviations 



71 +1 



1 



^ \p-2 



(97) 



^'1 \Pi 



and expanding (96), we get 



(98) 



«i 



+ e' 



1 



^'2 



hi 



» + l \ 



Ro ) 



= -(' 





logw 



0. 



n + r 



i?2 , 



log 



OJ 



= 0. 



Finally, expressing X„ and \i. in terms of pi and po, dividing by C'^, and 

 employing the arc lengths 



ds\ = e^ dv, ds-2 = c^ du 



along the w and v isothermal parameter curves, (98) becomes 



_a_A\ _ Ji(i\ _ J_ , J a^iogco) 



dSi\K2/ 



(99) 



3*^2 \'<l/ 



PlKi P2K2 



dsi dso 



= 0. 



The quantities pi, p2, i?i, R2, w in (99) are all geometric quantities, 

 and this equation expresses an intrinsic property of our " ?? " system, 

 for it is evidently true for any and every set of orthogonal isothermal 

 curves that may be chosen. Hence, 



Theorem 20. Property V. Construct any isoihcrmal net on the 

 surface. At any point 0, this net determines two orthogonal directions 

 in which there pa^s two isothermal curves of the net and two hyperoscu- 

 lating curves of Property III. If pi, p2, Ri, R2 are the radii of geodesic 

 curvature of these four curves, Si, S2, the arc lengths along the isothermal 

 curves, and 00, the tangent of the angle between the fixed direction of 

 Property III and the isothermal curve icith arc S2, then, as loe move along 

 the surf ace from 0, these quantities vary so as to satisfy the relation 



where 



d_ V 



dS2\K-ij 



1^ _ 

 Ki 



dS] \ Ko 



P\H\ p'lK-1 



d- (log g?) 

 dsi ds2 



0, 



'1 n -{- V 



CO -- 

 \Pi 



1 _ 1/1 _ !L±I 



K2 W\P2 R2 , 



