314 WILSON AND MOORE. 



If in four dimensions we use ^, o>, bra, Crs to avoid subscripts, we have 



= - |(aiX. + MlXr) + T](jJiX + /3lX,) + UrOi, (72') 



0), = - 2,c,,y(^' - vrl 



= — |(aoXr + M2Xr) + '^(lJL-2^r + P2>^r) — Vrt,. 



It is important to observe that the theory of the 2-surface in four 

 or more dimensions is not the same as the theory of the moving axes: 

 for the 2, or 7i — 2 normals, to the surface are to a large extent indeter- 

 minate so far as the surface itself is concerned. It is the set of ciuan- 

 tities V which render the normal system definite and upon which the 

 rate of change of the normal vectors depends as in the above equations. 

 The theory of the set of moving axes is a step further than the theory 

 of the surface and as far as the surface alone is concerned we may 

 disregard the v's so long as we do not need to differentiate the normals. 

 In this respect there is the same difference between surface theory 

 and the theory of moving axes (of which two are tangent to the surface) 

 as between the theory of a twisted curve in three dimensions and the 

 theory of moving ^axes of which only one is tangent to the curve. 

 If the differential theory of a curve is treated from the point of view 

 of the quadratic form (in one variable), the v which must be intro- 

 duced in the case of a twisted curve in three dimensions is related to 

 the radius of torsion. In the curve theory the set of axes is rendered 

 definite by assuming that the normal axes are along the principal 

 normal and binormal and if we desire to keep moving axes in our theory 

 of surfaces it will be desirable to specialize the normal axes in some 

 such way as in the case of curves in three dimensions. 



29. Tangent plane and normal space. Two elements which 

 have strictly to do with the surface alone are the tangent plane and 

 the normal plane or (w— 2)-space. Following the notation of Gibbs 

 (for the outer product) we may write the unit tangent plane and its 

 differential as 



M = |x-n, f/M = dixr\ + t<dT\. (73) 



The unit normal space is, 



N = i^xw or N = Z1XZ2X . . . xz„_2 , (73') 



dN = <^x(o + t^yidoi or dN = dzi^z-y^zix . . . xZn-2 -f etc. 



