1921-22.] Concomitants of Quadratic Differential Forms. 197 
where the forms of the functions P. Q, R, S, T are such as to allow one, 
and only one, eliminant when x-^_, x^, x.^ , x^ are made to disappear. When 
two neighbouring points are taken, represented by {x^, . . ., x^^ and 
{x-^-\-dx^, . . ., x^-[-dx^, the ‘‘distance” ds between them is given by 
ds^ = + r/Q2 + dW + dS2 + d.T‘^ 
= {a, b, c, d, f, g, h, I, m, n'^dx^ , dx ^ , dx ^ , dx^Y, 
with the preceding notation. The simplest case occurs when one of the 
original coordinates vanishes, say 
T = 0: 
the simplest expression in that event occurs by taking 
so that 
P — x^ 5 Q — X () , 1\ — i/Cg , & — , 
d.s- = dxY + dx^ + dxY + dx^ . 
In all cases, the variables x ^ , x.^ , x^, x^ are subject, or must be considered 
subject, to transformations of the most general kind; and we are therefore 
led to consider what functions of the coefficients a, . . ., in the differ- 
ential form are invariantive, and what is the geometrical significance of 
these invariantive functions. 
The analytical investigation of the invariantive forms has been effected 
in the preceding part of this paper. From the results obtained, especially 
in connection with the line-covariants, it appears that two of them, viz. 
^P), A(P), 
are of fundamental importance for the construction alike of line- co variants 
and of the differential invariants of the quadratic differential form which is 
the expression of ds^. These covariants, however, are only relative ; that is 
to say, when the variables are transformed, the same functions of the new 
coefficients and the new variables are respectively equal to the functions of 
the old coefficients and the old variables, save as to a factor which is a 
power of the modulus of transformation, the power of the modulus depend- 
ing on the particular covariant. 
The consideration of the simplest transformation 
x-^ = ax^t x^ = Yx^, x^ - yxY, x^ = 
shows that the covariantive quantity 
A(P) 
remains absolutely unaltered : it is therefore an absolute invariant : and, 
accordingly, it represents some geometrical property of the selected 
amplitude. We proceed to the interpretation. 
