326 WILSON AND MOORE. 



As 5y is arbitrary we infer that the condition for a minimal surface is 



dxi \dx2 ) dx2 \dxi / 



The equation further simphfies to 



dy dM dy ^ dM _ 

 dx2 dxi dxi dxo 



We may use (74) and (74') to modify the results to 



(|X2 + 11X2) • (ax-nXi + \i,xr\\^ - |ix|Xi _ px|Xi) 

 — (|Xi + "HXi) • (olxtWo + jJLxTiXa — [JLx|X2 — pxIXa) = 0. 



When we multiply the equation out we find 



(a + P) (X1X2 - X1X2) = 0. 



The term X1X2 — XiXo cannot vanish because it is equal to — Va 

 as may readily be shown from the defining relations of X and X; 



X1X2 - X1X2 = Va. (91) 



Hence the condition for a minimal surface is a + P = 0. Thus: 

 In any number of dimensions the condition for a minimal surface is that 

 the mean curvature shall vanish at each point of the surface}^ This is 

 the immediate generalization of the condition in three dimensions. 



By reference to (78) we see that for a minimal surface, dM'dM = 

 — Gds^. This relation in three dimensions is interpreted as showing 

 that the spherical representation of a minimal surface is conformal: 

 for dM'dM = dn*dn, n being a unit normal, and dn-dn is the 

 differential of arc in the spherical representation. In higher dimen- 

 sions we can merely say that: The magnitude of d'M./ds is the same 

 for all directions through a point on a minimal surface. 



Zl. The intersection of consecutive normals. Let N be the 

 unit normal space of n — 2 dimensions at any point of the surface, and 

 r a vector from that point. The equation of the normal space is 



36 This result has been stated by Levi, loc. cit. 



