SURFACES IN HYPERSPACE. 317 



30. Square of element of surface. Consider (fM • (/M which is 

 numerically equal to c/N-c/N. 



dm- dm = [ax-qsX.f/.r, + |Jix-nSM.r, - H^x|i:x,rfav - px|sMa;r]2. 

 Now (axT^).(ax-q) = a«a-n'T| — (a.r|)2 = a^, 



(ax-n) . (jjLx-q) = a.(i, (axii).(|xx|) = o, etc. 



dM-dM = a'^I,\r\dxrdxs + [i^^ZKXsdxrdxs + ['^-'EKKdxrdx, 



+ p2i:X,X.rf.ivrf.i-, + 2a.|xSX,Mavc/.Vs+2p.|xSX,Xsf/.rr(^.T,. 



By (69) we have \k-^ = a-^ - G. Hence 



dM'dM = — G[I,\rKdxrdxs + Xr^sd-Vrdxs] + a.p[z:XrX,rf.ivc?Xs 



+ XrXjC?.TrC?Xj] 

 + a-<il,\,\dxrdxs -\- P*PSXrXsrf.rrrf.Ts 

 + (a.|X + p-|A)[2X,X3(/.tvc?.r, + ZKXdxrdxs]. 



Now ars may be expressed in terms of the X's as bra was expressed in 

 (65'). Then, 



Qrs == ClX,.Xs + f'2(XAs + XAs) + CaXrXs . 



When the invariants ci, C2, f's are determined by means of (63!) we find 

 Ci = 1, C2 — 0, cs = 1. Hence 



Ors = X,Xs + XrX« • (77) 



dm 'dm = - G^Ursdxrdxs + (a + ^)-[aE\r\sdxrdxs + |J^2(XA3 



+ \rK)dXrdXa + pSXrXsrf.TrCfa's], 



or 



(/M-(/M = - 6V + (a + p)-^. (78) 



Hence: T/if? square of the differential of the tangent plane is equal to 

 the scalar product of the vector invariant O- + P and the vector second 

 fundamental forvi ^ 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 a, + P 

 and ^ are generally regarded as scalar quantities, dm-dm is replaced 

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



