MA THEM A TICS: WILSON AND MOORE 
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linear segment noncollinear with h, is a simple generalization of ordinary 
surfaces = 3) in that by a proper choice of parametric curves the 
first and second fundamental forms reduce simultaneously to sums of 
squares. The property is not invariant under a projective trans- 
formation. 
Lines upon surfaces. — Kommerell defines as principal directions those 
for which the normal curvature is a maximum or minimum, and sub- 
sequent authors have followed him. This gives four directions at 
each point, and they are not simply related to one another. We have 
chosen to give as a definition of principal directions those for which h 
and fjL are perpendicular. There are two such directions at each point, 
they are orthogonal and have the properties that the differential tangent 
planes in the two directions are perpendicular and the values of the 
rate of change dM/ds of the tangent plane is numerically a maximum 
or minimum. 
Kommerell defines asymptotic lines, and Levi follows him after re- 
stricting the definition to a special case. The definition is not sat- 
isfactory, and we define asymptotic lines as those for which h and a 
are perpendicular. The rate of turning of the tangent plane along these 
directions is the square root of the negative of the total curvature. The 
differential equations of the lines are h.^^ = 0. The lines are bisected 
by the principal directions. 
Our definitions for principal directions and asymptotic lines become 
illusory for minimum surfaces. 
Segre's characteristics exist when the indicatrix lies in a plane with 
the surface point O and are then those directions which make a tangent 
to the indicatrix. The asymptotic directions divide the characteristic 
directions harmonically. 
Development of surfaces. — In the neighborhood of a point a surface may 
usually be developed in either of two standard forms. 
For the first form h lies along the axis of Zi, the plane of the indicatrix is 
parallel to the axis of Zz, and the axis of x and y are along the principal 
directions. A hyperplane tangent to the surface at O will cut the sur- 
face in real, imaginary, or coincident directions according as it cuts the 
indicatrix in real, imaginary, or coincident points. The second standard 
form of the surface has the property that the two perpendicular hyper- 
23 = hM^'' - + 2 Bxy, 
or 22 = J {Ax'' H- 2 Bxy -F Cy"), 
Z2 = if(x,~f), 
Zi = 0,i > 3, 
Z2 = i Dx\ 23 = J Ey\ 
