199 
1921-22.] Concomitants of Quadratic Differential Forms. 
Aj 2 A , A 22 -r A , we momentarily allow c and d of the differential form 
not to vanish, and make n to vanish ; then 
An_ 
bed 
b 
A 
I. 
1 
ah - Id ’ 
"^12 _ 
- cdli 
h 
A 
cd{ab - b^) 
ab - Id ’ 
A 22 _ 
eda 
a 
A cd{ab - h^) ah - ’ 
in the limit. Hence the limiting form of F is 
1 = 1-^22 “ ^^h2“t ^hi) 
77 7 — iT7\l- ~ h{a-J}.^ — ae)h-^ — ^a^Jicy 26^72/2-1- 
\(ah - h") 
+ h{a^{2h^ - T>2) - + a{&2(27^2 - a^) - 7>i^}]. 
Also, the value of f is given by 
i^ah-jd. 
Now for the line-element associated with a surface 
adx-^^ 4- llidx^dx^ + hdx^ , 
the Gauss-measure of curvature of the surface is precisely this quantity 
_F 
f ■ 
Accordingly, Riemann takes the quantity 
A(P) ’ 
associated with the quantities P, which are compounded out of two 
directions , dy ^ , dy ^ , dy^ and dz ^ , dz.^ , dz ^ , dz^ in the four-fold 
manifold, to be a measure of the curvature of the manifold as associated 
with two directions. Denoting this measure of curvature by we have 
_K 
A(P)“ ’ 
as the Riemann expression for the curvature of the four-fold amplitude, 
connected with the two directions chosen at the place determined by the 
values of x^, x^, x^. 
38. This measure of curvature varies from one pair of directions to 
another. Where it is a maximum or a minimum or is stationary, we have 
^ = 0 , forr = l, 2, 3, 4, 5, 6 , 
1 &«(P) SA(P)^ 
and therefore 
