1921 -22.] Concomitants of Quadratic Differential Forms. 151 
5. The quantities dx ^ , dx ^ , dx^ , dx^ may be regarded as defining a 
direction at the position x^, x<^, x.^, x^ in the four-dimensional space ; to this 
direction there corresponds the point X^, Xo, Xg, X^. The line Pg 
then corresponds to a combination of any two directions at a position. The 
plane , Ug , Ug , corresponds to a combination of any three directions 
at a position. A combination of any four directions at a position gives a 
four-dimensional volume which is covariantive under the transformations 
considered. 
6. In the second place, and having regard to the occurrence of Gauss’s 
invariant for surfaces, we can expect to have invariants of similar 
character ; also, it may happen — indeed, it does happen — that concomitants 
of an entirely new character enter. For the purpose, it will be necessary 
to consider the derivatives of the coefficients a, . . ., n, and to consider 
their variations when the independent variables are transformed. There 
is nothing to impose a necessary limit upon the order of the derivatives 
retained ; so, remembering other investigations of a like nature and also 
the analysis used in Einstein’s memoirs, we shall retain derivatives of the 
coefficients in the differential quadratic form, of the first order, and of the 
second order. 
Thus, among the aggregate of invariantive functions associated with 
a differential quadratic form, we shall require 
(i) The customary concomitants (point-covariants, line-covariants, plane- 
covariants, in the different sets of variables) of the quaternary 
form, arising from the differential form when the differential ele- 
ments dx ^ , dx ^ , dx ^ , dx^ are subjected to linear transformation : 
(ii) The concomitants (whether pure differential invariants, or covariants 
of the various types involving derivatives of the coefficients) of 
the differential form. 
Method Adopted. 
7. In Einstein’s work,* and in the exposition of his work as given by 
others,]* the initial analysis is directed towards the construction of the 
conditions for the equivalence of the general form and the special form. 
* For convenience, I refer specially to the general account given by him in his memoir 
“ Die Grimdlage der allgemeinen Kelativitatstheorie,” Annalen der Physik, Bd. xlix (1916), 
pp. 769-822. 
t Particular mention should be made of three important and comprehensive accounts of 
various investigations connected with the general theory of relativity which have recently 
appeared, viz.; De Bonder, La gravifique einsteinienne (Gauthier-Villars, Paris, 1921); 
W. Pauli, jr., “ Relativitatstheorie,” Encycl. d. math. TViss., Bd. v, 2, Heft 4 (1921); 
Marcolongo, Relativitd (Principato, Messina, 1921). The amplest references are given. 
