148 Proceedings of the Royal Society of Edinburgh. [Sess. 
vocabulary of metaphysics has been introduced. My aim is mathematics, 
not metaphysics ; to me, an ‘‘ event ” is sufficiently indicated by the four 
variables x, y, z, t, subject to all classes of change of variables ; and my 
investigations do not quarrel with the definition of the ‘‘ interval ” between 
two “ events ” as given by 
ds^ = -dx^- dy"^ - dz^ + dt‘^, 
with the warning notice that the conservation of this measure of ds^ 
through all change does not specify the universe through all time, any 
more than the conservation of the Gauss surface-element ds^ would specify 
all surfaces having that element. 
For subsequent purposes, it proves convenient to change the variables 
by taking x, y, z, t as four independent functions of four independent 
variables x-^^, x^, x^, x^] and then ds^ becomes 
4 4 
ds^ = ^ ^ i Qmndx^dx^ , 
m~l n=\ 
with the limitation 
9mn ~ 9nm • 
But my intention is to discuss these forms without any necessary initial 
and persistent reference to the symbol measuring an event-interval ; and 
therefore a quadratic form 
a dx-^ + h dx^ + c dx^ + d dx^ 
- 1 - 2fdx^dx^ + 2g dx^dx-^ + 2h dx-^dx^ 
-f- 21 dx^dx^ -f 2m dx^dx^ -i- 2n dx^dx^ 
is postulated, sometimes to be written in the forms 
(«, 6, c, d, f, g, h, I, m, n ^ dx ^ , dx ^ , dx^ , dx^)'^, (a ^ dx)'^. 
The ten quantities a, b, c, d, f, g, h, I, m, n are called the coefficients of the 
form : in the general case, they are any functions of the variables ; in the 
particular Einstein case, they must be such functions of the variables as 
will make the preceding form equivalent to the interval-form. 
3. Now, when the variables x-^, x^, x^, x^duYe changed, there are con- 
sequent changes of distinct types. The four quantities dx ^ , dx ^ , dx ^ , dxi^ 
become homogeneous linear functions of dx(, dx^, dxff dx^, with coefficients 
that are functions of the variables involving no differential elements dx . 
The ten coefficients a give way to ten new coefficients a, according to 
the equivalence of the forms, as represented by the relation 
{a\ dxY = (ff dxY ; 
that is, the ten new coefficients a are homogeneous linear functions of 
