354 WILSON AND MOORE. 



just mentioned. The first, which follows Kommerell, gives four 

 directions on the surface which are not perpendicular and which in 

 the case of three dimensions reduce to the principal directions and the 

 asymptotic directions. We do not ordinarily associate common 

 properties to these two sets of directions. The second definition, sug- 

 gested above, also gives foiu* directions and in this case the reduction 

 is to the principal directions combined with their bisectors. We 

 do not usually investigate these directions or associate common prop- 

 erties to them and the principal directions. One great advantage 

 of the third generalization is that we have, as principal, two and only 

 two directions at each point and these directions are perpendicular. 

 The differential equations of the principal directions as defined by 

 h-ix = are, from (87), 



2„h-y„XWX(^) = 0. 

 By (63) and X- = liCirs^^^^ we may write 



Zrstii'Yrsar+U tX(')X(«)(- 1)(^+1) = 0, 



or 



^rsth-Yrsi- iy+'ar^i,td.vtd.vs = 0, (121) 



Written out at length we have 



h'[(yiia2i — y2ian)dxi^ -\- (ynCh2 — y22an)dxidx2 



+ (yi2022 — y-22aa)dx2^] = 0. 



This equation is similar to the ordinary equation except that y,s 

 replaces brs and the whole is multiplied by h. 



If the lines of curvature are taken as parametric lines, 



h*yiia2i = h-yoiou, h'yioa22 = h*y22a2i. 



These equations, since yna22 — y220ii 5^ 0, demand that 012 = and 

 h*y2i = 0. The condition that the lines of curvature he parametric is no 

 longer au = 0, yi2 = 0; the normal yi2 need merely be perpendicular to h. 

 In all this work h may be replaced by its value Srsfl'^Vrs if desired. 

 Special considerations need to be developed for the case h = 0. 



45. Asymptotic lines. When we seek for a generalization for 

 asymptotic lines we may consider the equation h*^ = 0, where ^ 

 is the second fundamental form, as defining asj-mptotic lines in general. 



