1921-22.] Concomitants of Quadratic Differential Forms. 
147 
XII. — The Concomitants (including Differential Invariants) of 
Quadratic Differential Forms in Four Variables. By Prof. 
A. R. Forsyth, Sc.D., F.R.S., Hon. F.R.S.E., Imperial College of 
Science and Technology, S.W. 
(MS. received March 20, 1922. Read May 8, 1922.) 
Statement of the Problem. 
I. The characteristic properties of quadratic differential forms, involving 
two or more independent variables, have been investigated from the days 
of Gauss onwards. Initially, the discussion arose for the case of two 
variables ; and, in its most general trend, it was concerned with a form 
ds^ = Edp^ + ^Edpdq + Gdq^, 
associated with surfaces, E, F, G being integral functions of p and q. 
But the relation does not, by itself, define a surface completely. When 
a surface is deformed in any manner, without stretching and without 
tearing, the quantity preserves its measure unchanged ; the measure 
is of fundamental importance. Consequently, the measure must remain 
unchanged whatever changes of the variables are admitted. Further, 
changes of the variables, of any kind, allow the existence of covariant 
concomitants which therefore persist through these changes. In particular, 
there is one function, of E, F, G and of their derivatives up to the second 
order inclusive, which persists unaltered; it is the Gauss measure of 
the curvature of the surface. 
The conservation of through all changes of variables (as well as 
that of its concomitant unvarying function) is not sufficient for a complete 
specification of the surface ; but it is a necessary preliminary condition. 
Similar remarks apply to ordinary space, when it is referred to triply 
orthogonal surfaces ; the investigations of Lame, Cayley, and Darboux 
are well known. 
2. In recent years, owing to the various stages in the Einstein theory 
of relativity, attention has been concentrated upon quadratic differential 
forms in four variables. Originally, the four variables were the ordinary 
space-coordinates, the Cartesian coordinates x, y, z or an equivalent set, 
and the time t. Later, some efforts have been made to use the four 
variables as specifying a four-dimensional space ; and not a little of the 
