SURFACES IN HYPERSPACE. 333 



The values of $0 and |Xx8 may be found in terms of y,, as follows. 

 From (93) and (94), 



$ = (hh - |i.p. - 55)a = ^yuy22 + ^y22yii - yi2y2i , (99) 



$2 = (K-xS |xx8 — 6xh 8xh — hx|JL hx|JL)a2 



= (- iy22xyiiy22xyii + |yiixyi2yi2xy22 + |yi2xy22ynxyi2), (100) 

 $3 = (iix5xh)2a3 = i(yuxyi2xy22)^ (lOl) 



where $2 and $3 represent the Gibbs's double products. ^^ Now 



$2xh = (|ii.x8 |JLx8xh)fl2, 2h = Sa("Vr« , 

 $2xh = (- la^'^-^y-yoxyn + ja("Viixyi2 + ia^^^^yi2xy22)(ynxyi2xy22). 



Choose, 



± a||xx8 = - ia(i2)y22xyii + ^a(")yiixyi2 + |a(22)yi2xy22 . 



Then, a|i.x8 [xxSxh = (- ^a^^-^y^-xyu + ia(^^'yuxyi2 + |a(22)yi2xy22) 



(- |a(i2)2 + ia(22)a(ii) + W^a(^^)y^,xyi2xy22, 



and the result checks. 



The double sign which arises here has come in through thie extraction 

 of a root. We may obtain from (87) the value of fAx8 as follows; 



|xx8 = ^^S,AMX(«)y„ X [2p,(X(p)X(«> - X(p)X{«')ypd • 



Li 



The coefficient of y22xyii is 



X(2)x(2)(X(i)' - X^)') - X(i)X(i)(X(2)2 - X(2)2), 



v/hich by virtue of (61') and (63) reduces to an/a,^ . The sign of the 

 term is therefore plus. In like manner the sign of p.x8xh may 

 be determined. Hence 



2alH-x8 = (ai2y22xyii + a22yiixyi2 + Ciiyi2xy22), 



2atp.x8xh = yiixyi2xy22 . (102) 



38 See Gibbs-Wilson, Vector Analysis, p. 306. As we are using the progres- 

 sive product 'i>3 = i *x*><* instead of I *x*»*' ^^^ ^^^ Wilson, Trans. 

 Conn. Acad., New Haven, 14, 1-57 (1908). 



