t 



366 WILSON AND MOORE. 



As M^ = 1, the first and third terms vanish; and as d'NL/du and 

 dM/dv are perpendicular to M, so must the spaces n>^d'M./du and 

 inx6M/8?) be perpendicular to M, and the other two terms will vanish. 

 Hence : The vector invariant 1 of the dyadic A vanishes. 



The invariant 3-vector S3 may be calculated. The work may be 

 simplified by taking the parameter curves orthogonal ■«dth u and v 

 equal to the arc along these curves (except for infinitesimals) in the 

 neighborhood of any preassigned point. 



Then m = |, n = "H and 



S3 =[n. (1.^)1 X'^-[|- (1x11)1 x!^^ 



as as 



= [n-(|xii)] X [ax-n + |x|i] - [|.(|x-n)] X [K-xn +|xp] 



= |xaxii + Tix|xp = - (|x-n) X (2h) = - 2]V[xh. 



Hence: The invariant ^-vector S3 = Ax i^ —2 Mxh, the space of the 

 tangent plane and the mean curvature, and of magnitude equal to the mean 

 curvature. 



Other invariants of the dyadic 



A = |(ax-n + |x|x) + Ti(|ixTi + |xp) 



are the dyadics A -Ac, Ac'A A:Ac, and so on, and the quantities 

 obtained from them by inserting dots and crosses. For instance, 



Ac-A = (ax-q + |x}jl) (ax| + |x|i.) + fiix-q + |xP) (jix-n + |xP), 

 and T4 = (Ac'A)x = - 2Mx[(a - P)xp,] = 4Mx|i,x5, 



A:Ac = 2(88 -{- \i\i - hh). 



Hence — M'Ss =- 2h, M'T4 = 4|Jix8, — AiA^ = 2$/a are the quan- 

 tities, found directly from the fundamental dyadic A, which have 

 been found of prime importance in the theory of surfaces.*^ 



49 The line of development here followed is the inverse of that which would 

 be followed in developing the surface theory from A. It is for brevity that 

 we choose merely to verify that the already known quantities h, jixS. and <I> of 

 surface theorv mav be derived from A. 



