350 WILSON AND MOORE. 



at the point the surface has the character of a three dimensional sur- 

 face which is not a developable. If this condition is satisfied identi- 

 cally the surface must lie in three dimensions. 



As an application of these results we may show that a minimum 

 ruled surface must always lie in three dimensions and consequently 

 be the helicoid. For as the surface is ruled the indicatrLx reduces to a 

 segment which, as h = 0, must pass through the surface point and 

 indeed have that point for raid point and hence the surface must lie 

 in three dimensions. 



44. Principal directions. If we take the value of a' from (89), 

 we find, as the condition that a' shall be maximum or minimum in 

 magnitude. 



= a'.rfa' = (h + HL sin20 + 5cos20) • (tJLcos20 - 5sin20) = a'-jx' 

 = h-K.cos2^ - h-8 sin20 + ([x^ - 62) sin2^ cos20 



+ (|x.8) (cos220 - sin22^). 



If we let X = tan^, the resulting equation in x is 



^4(|jL.6 - |i.h) + 2a-3(52 - |Jl2 - h-S) - 6a;2iJL.5 + 2a;(|x2 - 62 _ h-5) 



+ HL-(5xh) = 0. 



This is of the fourth degree and hence there are four directions of 

 maximum or minimum for the magnitude of the normal curvature 

 (Kommerell). Two of these directions must be real; for if we choose 

 |A and 8, which may be any radii of the indicatrix, as the axes of the 

 indicatrix, jx • 8 = 0, and the coefficients of the first and last terms are 

 opposite in sign. If a single pair of these four directions are orthog- 

 onal, it must be possible to choose jJL and 8 so that x = and x = 0° 

 satisfy the equations, that is, so that |A*8 ± |i'h= orJiL*8= |i.-h. 

 This means that: // tivo of the directions of maximum or minimum 

 normal curvature are perpendicular one of the axes of the indicatrix 

 must he perpendicular to h. If the four directions are perpendicular 

 in pairs x and — 1/x must satisfy the biquadratic and 



|i.6 - ii.h= |x-8 + (i-h, |jl2 _ 62 - h-8 = |j.2_ 62+ h-8 or 



jx-h = h-8 = 0. 



Hence: // the directions of maximum or minimum normal curvature 

 are perpendicular in pairs, the plane of the indicatrix must he perpendicu- 

 lar to h, that is, h must he along the axis of Cone I. 



