SURFACES IN HYPERSPACE. 333 



The values of $2 and |x.x8 may be found in terms of y,-,- as follows. 

 From (93) and (94), 



$ = (hh - K-K- - 55)a = ^yny-i-i + iy22yii - yi2y2i , (99) 



#2 = (p,x5 |xx5 — 5xh 5xh — hxji hxHi.)a- 



= (- iy22xyiiy22><yii + |ynxyi2yi2xy22 + ^yi2xy22yuxyi2), (lOO) 



$3 = (K.x5xh)2a3 = i(ynxyi2xy22)2, (101) 



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

 $2xh = (|J.x8 |JLx8xh)a2^ 2h = 2a("Vrs , 



*2xh = (- ia(l2)y22xyii + |a(">iixyi2 + ia(22)yi2xyo2)(yiiXyi2xy22). 



Choose, 



=t a^ix.x5 = - ia(i2)y22xyii + ia(ii)yuxyi2 + |a(22)yi2xy22 . 



Then, 01^x5 |i.x8xh = (- la^'^-^yo^xyn + ^a^i^yuxyio + ^a(22)yj2xy22) 



(- ia(i2)2 + ia(22)a(ii) + ia(ii^a(22))yuxyi2xy22, 



and the result checks. 



The double sign which arises here has come in through the extraction- 

 of a root. We may obtain from (87) the value of }JLx8 as follows; 



hx8 = ^^2„XWX(»)y,, X pp,(X(p)X(<'^ - X(p)X('?')yp,] . 



The coefficient of y22xyii is 



X(2)x(2)(X(i)' - X(>)') - X(i)XW(X(2)' - X(2)'), 



which by virtue of (61') and (63) reduces to aniah . The sign of the 

 term is therefore plus. In like manner, the sign of JJLx8xh may 

 be determined. Hence 



2afK'x5 = (ai2y22xyii + a22yiixyi2 + aiiyi2xy22), 



2aiK.x8xh = yiixyi2xy22 . (102) 



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

 sive product <l>3 = ^ <i>x*x* instead of ,} *x*»*- ^^e also Wilson, Trans. 

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



