SURFACES IN HYPERSPACE. 325 



The relation (69), that is, a«P — [i^ = G, maybe interpreted on 

 our indicatrix. For 



a. p = h- - 6^ h^ - (62 + |i2) = G. (90) 



Now the sum 82 -{- |i2 of the squares of two conjugate radii of an 

 eUipse is constant and equal to a- + h-, 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 sevii-axes 

 of the indicatrix}^ 



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

 face may be written as 



PJ.r,rf.r, = ^1- X — dx^dx^. 

 dxi dx2 



To find the condition for a minimal surface we write 



= 5 j C(^'-P)^dxidx2 = J j (^^ ^^i^-^2. 



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



^J/dT'Mdxidxo, 



or = X ^— X -—• 



OXi 0x2 0X2 OXi 



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



f f^-^X^- Mdx^x. = - f f 5y X ,^ (f-^ . m) dx^x., 

 J J dxi dx2 'J ^ 5.r, \a.To / 



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

 then 



II 



Sy X 



dxi \dx2 / dx2 \dXi, y 



dxidxo = 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. 

 Eisenhart, Ajner. J. Math., 34, 215-236 (1912), where references to earlier work 

 will be found. 



