SURFACES IN HYPERSPACE. 325 



The relation (69), that is, a-P — |x2 = (?, maybe interpreted on 

 our indicatrix. For 



a.p = h^ - 82, h? - (82 + jx2) = G. (90) 



Now the sum 82 -(- |x2 of the squares of two conjugate radii of an 

 eUipse is constant and equal to a- + b^, the sum of the squares of the 

 semi-axes. Hence: The Gaussian invariant G is the difference of the 

 square of the mean curvature and the sum of the squares of the semi-axes 

 of the indicatrix}'^ 



36. Minimal surfaces. ^^ The vector element of area of a sur- 

 face may be written as 



"Pdxidxo — — X — dxidxi. 

 dxi dx2 



To find the condition for a minimal surface we write 



= 5// (P«P)'t?x-i(/a-2 =11 7^-^ ^^*if^^2- 



If M is the unit tangent plane as heretofore, the condition becomes 



=y/6F-Mdxidx2, 



dxi dx2 dXi dxi 

 We have to integrate two terms by parts, one of which is 



'J ^ dxi dxo 'J 'J bx\ \dx2 / 



omitting the integrated term which vanishes at the limits; we have 

 then 



// 



5y X 



dxi \dx2 / dx2 \dxi , 



dxidx2 = 0. 



34 This result is stated by Levi, loc. cit., p. 71. 



35 For special developments on minimum surfaces see Levi, loc. cit., p. 90. 

 Ei.senhart, Amer. J. Math., 34, 215-2.36 (1912), where references to earlier work 

 will l)e found. 



