168 LIPKA. 



We may take for the equation of the conic of Property III lying in 

 the tangent plane at and passing through in the fixed direction w, 



(49) aof + ai^Tj + a2r?2+ SsHti - co^) = 0, 



where ao, ai, a2, co are functions of the coordinates (u, v) of only, and 

 ^, 7] are the coordinates of the center of geodesic curvature referred to 

 as origin. Substituting the values of ^, 17 as given by (40), we find 



„_ (/3o+/3i/+/32/2)(l + ^,'2) 



(50) V 



3(1 + cov ) 



where the /3's are functions of u, v only. Comparing this with (48) 

 we see that a and b are particular functions of u, v, v' satisfying the 

 condition 



(51) 



3/-fc(l+t'") 3(l+co«') 



Finall}^ combining (47) and (51), we may state 



Theorem 8. The viost general trijily infinite system of curves on a 

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

 equation of the form (47), in which 



[^o+/3it/+^2i/^][3/-6(l+/^)] , .. , . ,. 

 3(1 + co/) 



The differential equation involves one arbitrary function of u, v, v' and 

 four arbitrary functions of u, v. 



Equation (47) expanded takes the form 



(52) v"'=D^+Dtv"+bv"^ 



where Do and Di are special functions of u, v, v'. From Theorem 5 

 we infer that all systems with Properties I and III will also possess 

 Property II provided the function b has the form 



(53) b = -7^. 



y — CO 



Substituting this in (51), we find 



(54) ^^_7o+ 7^+72.'^ 



V — Oi 



where the 7's are arbitrary functions of u, v. The value of v" in (48) 

 corresponding to a hyperosculating curve is now given by 



