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MA THEM A TICS: WILSON AND MOORE 
change in any number of dimensions, could stand without repetition. 
Our own work therefore need only begin with the study of the second 
fundamental form. As, however, Ricci's absolute differential calculus 
is neither well-known nor published in particularly accessible form, it 
seemed best to prefix to our work an explanation of that calculus. 
The second fundamental form. — If y (w, v) be the vector which deter- 
mines the points on the surface, the second fundamental form is found to 
be a vector form 
^ = yiidxl + 2 yi2 dxidx2 + y22dxl = Zrs yrs dxrdxs, 
where yn, yn, y22 are the second covariant derivatives of y with respect 
to u and v and are vectors normal to the surface. If ds denote a differ- 
ential of arc and 
X(r) = dxr/ds, V^*-^ = dxr/ds', r = 1,2, 
be the differential equations of a system of curves X on the surface and 
of their orthogonal trajectories, the three vectors a, ^, /x, 
a = Zrs X^"-^ X^«) yrs, iS = 2rs X'^**) X'(«> yrs, 
fjL = X'(^) y^3 = 2r. X('-> X'(«> y„, 
are invariant of the parameter system on the surface and are associated 
with a direction X. Further 
y„ = a\r\s + (Xr X' + X,' \s) + /3 X' x', 
where Xr is the dual or reciprocal system to X^**^ defined by Xr = l^sdra X^«^ 
if ds'^ = 'Zrs drs dxr dxg bc the first fundamental form. 
The geometric interpretation of a is the normal curvature of the 
surface in the direction X (i.e., the curvature of the geodesic tangent to 
X); of /?, the same for the perpendicular direction; of jjl, the rate of 
change of a unit vector drawn in the surface perpendicular to the geo- 
desic tangent to X. The Gaussian or total curvature G is G = a-^ — ji^, 
the difference of the scalar product of a and jS and the square of jn; this 
is independent of the direction X and is one of the prime invariants of the 
surface. There is a vector h defined by the equation 
which is also independent of the direction X and which we call the mean 
(vector) curvature of the surface at the point considered; the magni- 
tude of h is another of the prime invariants of the surface. 
If M be a unit (vector) tangent plane to the surface, and dM its 
differential, the second fundamental (vector) form ^ may be written 
