SURFACE MOTION FOR ANY POSITIONAL FIELD OF FORCE. 181 



of the hyperosculating curves of Property III which have the directions 

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



(96) 



e^col 



'1 n-\-V 



Introducing the abbreviations 



CO \P2 



(97) 



\Pi Ri . 



1 



C0\P2 



log CO 



= 0. 



and expanding (96), we get 



-) - (-) 



(98) 



+ .' 





K2 





log CO 



0. 



Finally, expressing X„ and X^ in terms of pi and P2, dividing by e^\ and 

 employing the arc lengths 



dsi = e^ dv, ds-2 



du 



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



dS2\Ki/ 



(99) 



dSi\K2/ 



^ 1 a^iog co) _ Q 



P]K] P2K2 dSidS2 



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

 and this equation expresses an intrinsic property of our " n " 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 isothermal 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 p\, po, Ri, R2 (ire the radii of geodesic 

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

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

 Property III and the isothermal curve with arc S2, then, as we move along 

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



dS2\li], 



A/i 



ds\K2/ 



"■piKl P2K-2 



where 



1 



— = CO 



Kl \Pl 



^1 . 



1^ 



K2 



C0\P2 



d"^ (log co) 

 dsi ds2 



i?2 



= 0, 



