211 
1921-22.] Concomitants of Quadratic Differential Forms. 
Now take a new variable r, such that 
r 
then 
adx-^ = dx^' = 
ihndS 
so that 
Ih. 
= 8*'’ 
72 L 2 
a^dx,^ ^ -^dr^ 
We have 
IVd 
dr^ 
^=(/777)^S/-^=. ±lr, 
where the doubtful sign is immaterial. 
Gathering the results together, we have 
d^^^ ^ - Mx^ - ^Mx^ - fdx^ + IHx^ 
mfi 
Jy2 _ ^2^.2^^^2 _ ^.2 g^j^2 _j_ 1 c ^dx^ + ( — 2 
^mli^ 
r J 
dx,‘^ 
as the expression giving the interval. 
44. The expression has been obtained merely by direct integration 
of the initial differential equations ; but, as yet, no conditions have been 
imposed. Now let the requirement be imposed that, as r tends to become 
infinite (the equivalent of no gravitation field), the expression for ds^ 
should degenerate to 
— dx"^ — djj^ — dz^ + df^ , 
or, in polar coordinates, 
— dx‘^ — r'^dO'^ — 7'^ sin^ ddcf)^ + dt‘^ . 
r = r, 
x^ = 6 ^ 
x^ — i J 
h=l , 
^^=1, 
k' = 0, 
Then we have 
