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. 



X„ 1 



r^' P. 





j_ _ 



Pi C" Pi c 



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 



111 



72 + X,. + 3X„co 



1 _ 7o — X„co — 3X„ 

 ^ 3? ' 



Now (66) may be written 



(67) (72 + X. + 3X.C.),, - (X.CO). + h - ^^' - ^"") + (^) 



— (log ui)uv = 0. 



Introducing the values of pi, p-:, Ri, Ro, this becomes 



03 \P2 i?2/ 



Introducing the abbreviations 



1 /I 3\ 1 1/1 3^ 



Kl \Pl til/ l^i W \P2 /i2y 



and expanding (68), we get 



(68) 



'''' (p-. - i 



— (logco)„„ = 0. 



(70) 



- 



\Kl/ u 



yK2 V 



+ 



X » X i 



Kl K-2 



— [log u]uv = 0. 



Finally, expressing X„ and X,. in terms of pi and po, dividing by c-\ and 

 remembering that the arc lengths along the ?/ and v isothermal para- 

 meter curves are given by 



dsi = e^dv, ds2 = e^du 



we may ■wTite (70) in the form 



(71) ±(1\-±(L]-JL-^JL- d'(\ogo^) ^ 



5^2 \lij dSi \K2/ piKi P2K2 dsi dso 



The quantities pi, P2, Ri, R-2, w entering (71) are all geometric quan- 

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



