340 AVILSON AND MOORE. 



It is seen from these equations that: The mean curvature h of a 

 ruled surface is one-half the normal component of the curvature i" 

 of an orthogonal trajectory of the rulings; the indicatrix is a conic of 

 which a pair of conjugate radii are the mean curvature h and the line 8 

 which is the normal component of g' , which gives the rate of turning of 

 the ridings; the total curvature of the surface is the negative of the square 

 of the normal component of g' . The rided surface is a special type 

 under the four-dimensional type in that the indicatrix passes through 

 the surface point considered. As the inverse of the pedal of an ellipse 

 with respect to a point on the ellipse is a parabola, the locus of points 

 where consecutive normal planes (spaces) meet a given normal plane 

 (space) is a parabola (parabolic cylinder) with its axis parallel to h. 



41. Developable surfaces. One particular parametric form for 

 a general surface, 



X = X, y = y, Zi = Zi(x, y), 



which expresses the surface as the intersection oi n — 2 cylinders 

 Zi = Zi(x, y), is often useful. In this case the vector coordinates 

 of the surface and the differential element of arc are 



p = a;i4- ?/j + Sziki, 



^p ^ (i _|_ 2 ^^ k .),/., + ( j + 2 ^^ ^,)dy 

 ax ay 



dp 'dp = 



-='i; 



dx dy 



Let Pi, qi be the derivatives of Zi with respect to x and y. Then, 



m = i + UpdHi, n = j + Zgiki. " (107') 



40 The actual determination of a possible set of values for ft and 8 maj' be 

 made when the values of \i.\i. + 88 and (axS are known. In this particular case 

 ^x8 = - icxd where c = f" - g'.f'g, d = g' - f'.g'f and h = kj. 

 Then since o = 1, fifi + 88 = hh = Jcc + dd. If n = ex and 8 = d/x, 

 then jin -f 88 = j.r-cc + dd/.r-, and x must be imity provided c and d have 

 distinct directions as they must have since H'x8 9^ 0. 



