SURFACES IN HYPERSPACE. 317 



30. Square of element of surface. Consider f/M-f/M which is 

 numerically equal to c?N*(/N. 



dM'dM= [axti2X.(/.iv + IxxiiSX.dr, - |ix|2X,c/a,v - ^^tZKdxrf. 



Now (axil) . (ax-q) = a-aTi-ii - (a.ii)2 = a2, 



(ax-n) . (jjLxTi) = a-fi, (ax-n).(|ix|) = o, etc. 



dM'dM = a'^^Wdxrdxs + \y^'L\r\sdxrdxs + [t^XWdxrdxs 



+ P-Z;X,X,f/.r,f/.rs + 2a'\i.^\rKdXrdxs -{- 2^'[i7:\r\sdxrdxs. 



By (69) we have |i2 = a- p - G. Hence 



dM'dM = — G['2\rKdxrdxs -{- Xr'^sdxrdx^] + a«p[2XrXsC?.xvc^a:g 



+ \ry^sdXrdXa] 



4- a«aSXrXsrfx,.c?a-s + p-PSXAsC^avc^-Ta 



+ (a.|x -f- p«li.)[2XrXsC?.rrC?.rs + ^K^dxrdxs]. 



Now Ors may be expressed in terms of the X's as brs was expressed in 

 (65'). Then, 



Clrs = ClX,Xs + 02(X,-Xs + XrXs) + ("3X^X3 • 



When the invariants ci, co, cs are determined by means of (65) we find 

 C'l = 1, c-2 = 0, C3 = 1. Hence 



ttrs = KK + XrX, . (77) 



dM'dM - - GZursdxrdxs + (a + p)-[a2XA,(^.rrC?.r, + 1^2 (X,X, 



+ \r\)dXrdXs + ^^XSsdXrdXs], 



or 



dM'dM = - GV + (a + P)-^. (78) 



Hence: The square of the differential of the tangent plane is equal to 

 the scalar yroduci of the vector invariant cl -(- P and the vector second 

 fundamental form ^ less the product of the Gaussian invariant G and 

 the first fundamental form <p. 



This relation holds also in three dimensions : but in this case ct + P 

 and ■^ are generally regarded as scalar quantities, dM • dM is replaced 

 by the square of the differential of the normal, — and, furthermore, 



