180 



LiPie\. 



times the geodesic curvature of the hyper osculating curve ivhich passes 

 through the point in the direction of the force vector. 



We may now show that the most general triply infinite system of 

 curves possessing Properties I, II', III, defined by the differential 

 equation (89), will also possess Property IV', where a fixed direction co 

 replaces the direction of the force vector, provided the condition 



(92) To + 7i CO + 72"- = (wu + cocoj + ?i(X„ — X„co) (1 + or) 



is satisfied. 



It is evident that Properties I, II', III, IV do not completely char- 

 acterize an " 11 " system. To complete the characterization, let us 

 compare the differential equations (89) and (82); we evidently have 



(93) CO = -; To=— + wX,; 7i = + ?i(X,,j - X„); 



(p (p (p (p (p 



72 = - -- - nhu - 



As in §8, conditions (93) may be reduced to (92) and 



^7o — n\,: — Wu\ 



(94) 



72 + ?iX„co I + 



= 0. 



CO 



In order that an eqviation of the form (89) should represent an " n " 

 system, it is, therefore, necessary and sufficient that the four arbitrary 

 functions 70, 71, 72, co satisfy (92) and (94). 



To interpret (94) geometrically, we may write it in the form 



(95) 



72 + X,+ (?i + l)X„a) — 



Introducing here the geodesic curvatures 



X« 1 



Xj, 



i - __ _ = _ 



Pi e^ p2 e" 



of the isothermal u and v parameter curves, and the geodesic curvatures 



1 _ 72 + Xr + (n + 1) X^ J_ _ 70 — {n + 1)X, — X»cj 

 Ra -(1 -\-n)e^o: ' R2 ~ (1 + n)e^ ' 



