SURFACES IN HYPERS PACE. 315 



In terms of the notation introduced above we have, from (71), 



dt = ^rtrdXr = d^Xd.l'r + \i-^r^rdXr + T\'^r'PrdXr , 

 dy\ = lir'^rdXr = [l>-^r\dXr + P2rXrf/.rr — ^'^rfrdXr . 



Hence 



dm = ax-qsXrc/av + [xxilSXrdrr - [t.xi'SKdxr - Px|SXr(fer. (74) 

 Now dy = Zyrdxr , | = 2X('-)yr , y\ = SX ('■)y, . 



The last two equations may be solved by inspection as 



7r = |Xr + "nXr , 



dy = %^\rdxr + r0Krdxr . (74') 



Hence 



dyxdM = t^CLxT\1,\rdXr'2:\rdXr + i>^\l.xy\I,\rdXrI:\rdXr 

 — T\X\l.xil,\rdXr7:\rdXr — y\^^>^i'2\rdXr'S,\rdXr 

 • = — ^>^'t\^{CL^rsWdXrdXs -{- ^lirs^r^sdXrdXs 



+ K.S„(XrX3 + \r\)dxrdx,} 



or dyxdM = — M^'^rsyrsd-Trdxs = — Mx'"!'. (75) 



This expression may be solved for ^ by multiplying by M. Thus,^^ 



M-(rfyxdM) = - M-(Mx^). (75') 



29 We shall use as a definition of the inner product that due to G. N. Lewis 

 (loc. cit., note 15) which has the advantage over the inner product of Grass- 

 mann that it is commutative. The interpretation of the inner product of 

 a p-dimensional parallelepiped and a g-dimensional parallelepiped where 

 q > p is a (q — p) -dimensional parallelepiped in the g-space perpendicular to 

 the p-space. The rules of operation with inner products have been developed 

 for a non-EucUdean case by Wilson and Lewis (loc. cit., note 15) and the rules 

 for the Euclidean case are not different except for an occasional change of 

 sign. As the product is distributive the rules may all be verified on or derived 

 from products of unit vectors. (For the transformation used in the text at 

 this point see Wilson and Lewis, p. 439). One of the most important rules is 

 that represented by such expansions as, 



(mxn) • (pxqxr) = (pxqxr) • (mxn) = 



(mxn) • (qxr)p + (mxn) • (rxp)q + (mxn) • (pxq)r. 



The general rule is to take from the larger factor as many of its factors as there 

 are factors in the smaller factor to form with them a scalar product, taking all 



