330 WILSON AND MOORE. 



As yiiy22 5^ y22yu this dyadic is not self conjugate. 



Q = [aXi2 + 21JLX1X1 + p\i2] [aXo2 + 21XX2X2 + PW] 

 - [ctXiX. + |i.(XiX2 + X2X1) + PX1X2]- 

 = [— |i.|Ji(XiX2 — X2X1) + aPXiX2 — PaXiX2 



+ (a|x - jxajXiX. + (H-p - Ph^)XiX2](XiX2 - X2X1). 



The first term is self -con jugate and the last two are anti-self-conjugate. 



i(^ + ^c) = [Kap + Pa) - Ji|x](XiX2 - XoXi)^ 

 = [hh - M-fA - 55]a, 



by (91). We find therefore that: The selfconjugate part of the vector 

 matrix Q, is the dyadic $ ichich defines Cone II, with the multiplier 

 c = a. We shall use for 4> the value 



$ = [hh - |XHi - 88]a, Cone II, (94) 



including the multiplier a; and for $-^ 



$-1 = [h'h' - fx>' - 5'5']a-i, Cone I. (94') 



The value of the scalar invariant fis of the dyadic fi and of the self- 

 conjugate part of 12 are the same. Hence, 



fis = <l's = yu-y22 - y,2' = (h^ - i^^ - 82)a = Ga. (95) 



We have therefore the result that: The Gaussian curvature G is 



a a 



the quotient of the scalar of the matrix of the second fundamental 

 form by the discriminant of the first fundamental form, in complete 

 analogy with the result in three dimensions which expresses G as the 

 quotient of the determinants of the two fundamental forms. It has 

 already been seen that the mean curvature h is expressed as 



2h = :2,-ya"%,-; 



