172 



LIPKA. 



Consider, at a point 0, the isothermal u and v parameter curves and 

 the hyperosculating curves of the system in these directions. Noting 

 that v" = for the isothermal curves, we have for the geodesic 

 curvatures of the u and v parameter curves, 



1 = _ ^. 1 = _ ^- 



Pi 



c^ ' Pi 



Again, for the hyperosculating curves, v" is given by (55), and the 

 geodesic curvatures of these curves in the directions of the parameter 

 curves are 



^1 



72 + X' + 3X„co 



-3e^ 



CO 



R2 



7o — X^cj — 3\v 



Now (66) may be written 



(67) (72 + Xr + 3X„aj)„ — (X„w)„ + 



^7o — 3X^ — X„w 



CO 



V \^ / V 



(log co)„„ = 0. 



Introducing the values of pi, po, Ri, R2, this becomes 



(68) 



e^co 



\Pi 





CO \p2 



A 



R2/ 



— (logco)„^ = 0. 



Introducing the abbreviations 

 (69) 



1 



Ki \pi lil/ K2 



CO \P2 



A 

 R2, 



and expanding (68), we get 

 (70) ' '^^ ^' 



vKl/ 



^^'2/ 



+ 



Xm 



Xj, 



K2 



— [log co]„^ = 0. 



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

 I'emembering that the arc lengths along the u and v isothermal para- 

 meter curves are given b\' 



dsi = e^dv, ds2 = e^du 



we may write (70) in the form 



(71) A/i\_±/i)_^ + J-_«-!M^ = o. 



dS2 \/C]/ dS\ \K2/ PlKl P2/C2 "^^l ^^2 



The quantities pi, p2, Ri, R2, co entering (71) are all geometric quan- 

 tities, and (71) expresses a relation connecting their rates of variation 



