316 WILSON AND MOORE. 



But C-(bxA) = (C-A)b + (b-C)-A, 



Here, Mv/M = and M«^ = 0. Hence 



{dy'M)'dM = - ^. (76) 



The vector dy-Jf/l is a 1 -vector in M perpendicular to dy and 

 ((/yM)'c?M is a 1-vector in c/M perpendicular to dyM. 



The expressions (75') or (76) hold of course in three dimensions as 

 the work by which they were obtained is independent of the number 

 of dimensions, greater than two. In ordinary surface theory we have 



^ = llrsbrsdxrd^s = " '^hdijhd^ = — dy ' clt,, 



where f^ are the direction cosines of the normal ^. If we multiply 

 this by 'tj to make a vector form we have 



^ = l^rsbrsdxrdxs = - Kdydl) = " dl'H^dy). 



The form is expressed in terms of the normal and its differential instead 

 of in terms of the tangent plane and its differential. We may make 

 the change by taking complements, ^° 



[dl'il^dy)]** = - [dl^il'^dy)*]* = - [dl^idy-M)]* = {dy'M)'dM. 



We have therefore arrived at a formula ^ = — (f/yM)'(ZM for the 

 (vector) second fundamental form which is the immediate generalization 

 of the formula in three dimensions. 



If we desire to express the second fundamental form in terms of the 

 normal (??— 2)-space N instead of in terms of M we can do so. 



possible combinations and adding with due regard to sign. For the case in 

 which the two factors are of equal order we have 



(inxn)«(pxq) = 



in-p n-p 

 in«q n«q 



These rules for obvious reasons are similar to those for regressive or mixed 

 products and the rule quoted at this point in the text is like Sltiller's theorems 

 (see Whitehead, Universal Algebra, p. 192). The complement, denoted by *, 

 which is used below is similar to Grassmann's supplement, except possibly for 

 sign. 



30 See Wilson and Lewis (loc. cit., note 15), p. 435. 



