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) ao^2+ ai^Tj + a2T+ ZeHl - «^) = 0, 



where ao, ai, ao, w 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 ^, rj as given by (40), we find 



^^0) ^ = ^oT^) ' 



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

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

 condition 



a-(X„+Xy) _ (^o+/3i/+/32/-) 

 ^^^ 3/- 5(1+/ 2) 3(l+coi-') ' 



Finally, combining (47) and (51), we may state 



Theorem 8. The vwst general triply infinite system of curves on a 

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

 equation of the form (47), in xehich 



[/ 3o+^y+/32/-^][3^/-5(l+/^)] , .. , . ,^ 



a = + (A»+ A.'' )• 



3(1 + cor ) 



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

 four arbitrary functions of u, v. 



Equation (47) expanded takes the form 



(52) v"'= Do^D,v"+bv"'- 



where Do and Di are special functions of ?/, 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 = ~j^. 



V — CO • 



Substituting this in (51), we find 



/^^x „ _ 70+ 7i^''+ 72^;' ^ 



a = — 



r — u> 



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

 corresponding to a hyperosculating curve is now given by 



