340 



WILSON AND MOORE. 



Moreover,*^ 



$g = G' = - (g' 



^8 = g'- r-g'i', 



- f'g'f'f = - 62. 



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

 ruled surface is one-half the 7iormal component of the curvature f" 

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

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

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

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

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

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

 the surface jwint 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), 



wliich 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 = .ri+ yj + ^Ziki, 



cZp = (i + 2 ^' kMv + (j + S ^' iii)dy, 

 dx dy 



dp -dp = 



1 + 2 



. dx 



dx'~ + 2^^^^-^dxdy + 

 dx dy 



1 + 2 



dziV 



(107) 



df 



Let pi, Qi be the derivatives of Zj with respect to x and y. Then, 



m = i + '^piki, n = j + 2giki. 



(107') 



40 The actual determination of a possible set of values for \i. and 8 may be 

 made when the values of jifi + 88 and \ixS are known. In this particular case 

 |xx8 = - icxd where c = f" - g'.f'g, d = g' - f'.g'f and h = ic. 

 Then since a = 1, jxp, + 88 = hh = |cc + dd. If ji. = ex and 8 = d/x, 

 then fifji + 88 = |.r'-cc + dd/.r-, and .r must be unity provided c and d have 

 distinct directions as they must have since nx8 ^ 0. 



