686 MOORE. 



The condition that the two directions be orthogonal is 



(52) 7niWuH3/i • rY + W2W2H J/2 • r)^ + 711^3' (.¥3 • rY + 0. 



This defines a plane of directions perpendicular to 7/ii^, W2^, vis^. 



From (48) we see that a linear relation among the m's gives a plane 

 of directions through the point and from (50) we see that the end of 

 the curvature vector will describe a conic in the curvature triangle 



^ , . , , ,, • , Mv(Mvr) M,-(M,-r) Mr(Mrr) . 



determmed by the pomts, ——- — -^, — — — — , — — — -— smce 



[Mi-ry [M-2-r)- [Ms-rY 



the substitution of (52) in (50) gives a quadratic relation in mu 1112, mz. 

 To simplify the work let 



(MvrY ''"' (M^-rY ^' Qh-rY ^ 



mi2(il/i-r)2 = X2, mi'(M2-rY = fj:\ mz'{ih-rY= i'- 



Then (50) takes the form 



(53) C = ^'•^' + ^'^ + '"'. 



A- + M" + V- 

 Then a linear relation 



(54) a-\ -\- bfi + cv = 



is equivalent to saying that the direction X, fx, v is perpendicular to 

 the direction defined by 



(55) ^ ^ ' 



V(MrrY' V{MrrY' V(Mz-rY 



From (53) and (54) the curvature of the directions perpendicular to 

 (55) is 



/ggx ^ ^ c--(X^.r + fxhj) + («X + bf^Yz 



c2(X2 + ^2) + (aX + 6m)2 



This is a conic and as seen before it must lie inside the triangle de- 

 termined by .T, y, z and must therefore be an ellipse. The sides of 

 the triangle are 



K7^ , _ X-.r + fi-y Vx + v'Z pry + vh 



X'' -\- IJL~ X" + V- fJL- -\- V- 



