364 WILSON AND MOORE. 



with the condition Zee = «"• From this it follows that 



Zl + Ki _ 22+ Kj ^ ^ Zn-1 + Kn-2 



Cl C2 C„_2 



The curve therefore lies in a plane through the .r axis and some line 

 in the z space; it is the common catenary and the result is: The only 

 minimal surface of revolution is the ordinary catenoid.^'^ 



As ai2 = and yi2 = 0, |Ax5 = 0. All surfaces of revolution are 

 of the type for which the indicatrix reduces to a linear segment. Our 

 lines of curvature coincide with Segre's characteristics and both lie 

 along the circles and the various directions assumed by the directrix 

 in the revolution. 



49, Note on a vectorial method of treating surfaces. 

 Another general method of dealing with the theory of surfaces 

 upon a vector basis may be mentioned without going much into 

 details. In the ordinary three dimensional case we set up the linear 

 vector function $ which expresses the differential normal dn in terms 

 of the displacement dr, i. e., c/n = c/r'$. As the properties of dyadics 

 $ are well knowni many properties of the surface may be found at 



48 



once. 



In the general case the tangent plane f/M is connected linearly with 

 the displacement dr. In fact if cZr = m du -\- ndv, a. differentiating 

 operator 



(125) 



may be written down which is invariant under a change of parameters. 

 (This is obvious since 



r\ r\ ^ r^ 



c?r«V = m'(n«M)c?u H' {ra-Wjdv — = du \- dv — , 



du dv du dv 



47 A geometric proof may be given as follows. Since the surface is minim nTn 

 h. = 0, and since the surface is of revolution the indicatrix reduces to a segment 

 (see below). Consequently the minimum surface of revolution is one for 

 which every point is axial with the center of the indicatrix (not its end) at the 

 surface point. Hence the minimum surface of revolution must lie in three 

 dimensions and in this case the surface is known to be a catenoid. 



48 For a brief discussion see Gibbs- Wilson, Vector Analysis, p. 411 ff. 



