202 
Proceedings of the Royal Society of Edinburgh. [Sess. 
Now, it is a known proposition * in the theory of homogeneous forms 
that the necessary and sufficient conditions securing the transformation of 
one form into another are that the same co variants and invariants vanish 
for the two sets of concomitants and that the absolute invariants are equal. 
When we have to deal with a form 
there is no absolute invariant under linear transformations. And we 
have seen that, in order to make all the line-covariants except A(P) vanish, 
it is necessary and sufficient that the relations 
D = 0, M = J=0, A = 0 
should be satisfied. 
We therefore infer that, if a form 
(a, h, c, d,f, g, h, /, m, n'^dx, dy, dz, dt)^ 
is to be transform ably equivalent to a form 
dx^^ + dx^ + dx^^ + dx^ , 
it is necessary that the twenty-one relations 
D=.0, M = 0, . . ., J = 0, A = 0 
(equivalent to only twenty independent relations) shall be satisfied ; and 
these relations are also sufficient, so far as concerns the mere alo^ebraic 
transformations of dx-^ , dx^, dx^, dx^. Further, under Lie’s theory of 
continuous groups in connection with infinitesimal transformations, we 
can now infer that the coefficients of ail forms equivalent, under any 
transformations whatever, to the form 
dx^ -j- dx^ -4- dx<^ dX/^ , 
must satisfy the twenty-one partial differential equations 
M=0, . . J = 0, A = 0, 
which are connected by a single linear relation. 
40. In the second example, we take the amplitude 
/{P, Q, K, S, T) = 0 
to be a space of constant measure of curvature W. Then we must have 
^(P) 
A(P) 
^W, 
* Aronliold, Crelle, t. Ixii (1863), pp. 281-345 ; Gram, Math. Ann., t. vii (1874)^ 
pp. 230-240. 
