196 Proceedings of the Royal Society of Edinburgh. [Sess. 
is a new line-covariant ; it will therefore be retained. Qt may be added 
that 
because of relations 
nd 4 - sm -f- tl = 0 , 
six in number, 
vk -t- ok -f- fu -t- re qnn- pi = 0 , 
fifteen in number, and 
ad -h qt -h sp + Ih + Jm + n2 = A , 
three in number; thus D'^f(P) would furnish no new constituent.) 
Hence by the association of D'r(P) with the lyreceding concomitants, we 
have an algebraically complete aggregate of independent concomitants of 
the quadratic form. 
This aggregate can be modified by the substitution of other con- 
comitants for those actually included ; the foregoing seem the simplest 
set to select. Other concomitants can easily be written down ; thus 
we could have 
dpviV), dj,w{F), dj,x(F), dp/{F), d^z(F), 
dfu(F), dfv(F), dfw{F), dfx(F), dfy{F), 8fz{F), 
where is the operator obtained in § 28. Again, we could have 
Dzv{F), Dx{F), By{F), D.(P) , 
D2?^(P), D‘^x{F), Bhj{F), Dh(F) 
D^x{F), D3^(P), D%(P) 
D^P), DMP), D’MP), 
and so on. But, as already stated, apparently the simplest aggregate has 
been selected. 
Four-Dimensional Space ; Riemann’s Measure of Curvature. 
36. Consider a five-dimensional space, for which the non-homogeneous 
coordinates are P, Q, R, S, T. Any four-dimensional amplitude in that 
space can be represented by a relation 
/(P, Q, R, S, T) = 0, 
= P(^i , 
, 
x^ , 
Q = 
= Q(^i , 
X'2 5 
^4) 
R^ 
= R(:Tj , 
X^ , 
^4) r 
S = 
= S(a-i, 
X^ , 
^3 , 
^4) 
T = 
X^ , 
X 3 , 
or by a set of relations 
