1921-22.] Concomitants of Quadratic Differential Forms. 149 
the ten old coefficients, the quantities entering into the linear expressions 
being functions of the variables. 
Two dissimilar considerations show that other magnitudes must be 
introduced. 
4 . In the first place, as dx -^ , dx ^ , dx ^^ , dx^ are homogeneous linear 
functions of dx^, dx^, dx^, dx^, we write 
dx^ = Xj , = X2 , dx^ = X 3 , dx^ = X4 , 
and regard , X2 , X3 , X4 as homogeneous coordinates of a point in 
ordinary space ; and we shall then have a quaternary form 
(f/, C, d, /”, 771, 71 ^ X 4 , X 2 , X 3 , X^i^, 
the variables X in which are subject to linear transformation. We are 
therefore led to consider the concomitants of such a quaternary form ; and, 
in doing so, we are bound to take account of two other sets of variables, 
viz. plane-variables and line-variables, in addition to point-variables. 
The four plane-variables Uj , U 2 , Ug , can be defined in either of two 
ways : (i) by the relation 
2,u,x,=2v'x/, 
r=l r=l 
or (ii) by the equations 
U,, XJ,, U3, u,= 
Y3, 
Y4 
Zi, 
Z2 j 
Z3 > 
Z4 
Ti, 
T2, 
T3, 
T4 
where the sets of variables Y, Z, T are subject to the same transformations 
as the variables X, that is, are cogredient with the variables X. The plane- 
variables U are contragredient with the variables X. 
The six line-variables Pj , Pg , Pg , P4 , P5 , Pg can be defined in either 
of two ways : (i) by the equations 
~ Q 2^3 ~ J ^6 ” Qi ^4 “ Q 4^1 » 
P2 ~ Qa^l ~ Qlfi '3 ) ^5 ~ Q 2^4 ~ Q 4^2 ’ 
P3 = Q1P2 — Q2^I ’ ^4 “ Qs ^4 ~ Q 4^3 ’ 
where the two sets of variables Q and R are cogredient with the variables 
X ; or (ii) by the equations 
Pi = W4 - V4 W4 , P, = V2W3 - YgWg , 
P2 = V2 W4 - V4W2 , P5 = VgW4 - V4 Wg , 
P3 = V3W4 - V4W3 , P4 = V4 W2 - V2W4 , 
where the two sets of variables V and W are cogredient with the variables 
