198 Proceedings of the Royal Society of Edinburgh. [Sess. 
37 . These variables Pj , . . Pg have been considered as line-variables 
from the point of view of convenience, just as , X2 , X3 , X^ have been 
considered as point- variables ; and the adopted definitions are 
P,==Y2Z3-Y3Z2, Pg = Y,Z,-YA, 
P2 = Y3Z,-Y/.3, Pg^YA-Y.Z^, 
P 3 = Y,Z 3 -Y 2 Z,, P, = Y 3 Z,-YA- 
To interpret the covariantive quantity in question, imagine a linear trans- 
formation of the cogredient sets of variables to be effected such that P3 
does not vanish while Pj , P^ , P4 , P5 , Pg vanish, thus of course securing 
the relation 
In these circumstances, ^dP) becomes FPp and A(P) becomes fP^^; and the 
quotient is an absolute invariant. Hence 
w(P)_F 
A(P)“f • 
But in order that the quantities P^ , P2 , P4 , P5 , Pg may vanish, while P3 
does not vanish, we must have 
Y3 = 0, Y4==0; Z3 = 0, Z4 = 0. 
We must now return to the differential form. The point-variables 
X4 , . . X4 are displacements dx -^ , dx ^ , dx ^ , dx^ in our amplitude of four- 
dimensions; and likewise for the sets Y^ , . . ., Y ^{ = dy-^, dy^, dy^, dy^y 
and Z4 , . . ., Z4 ( = dz -^ , dz ^ , dz ^ , dz^. Accordingly, for the purposes of 
the special variables P, we are considering two displacements dy ^ , dy^ 
when ^3 and y^ do not vary, and dz -^ , dz^ when 03 and 2^4 do not vary ; 
while for the general variables P, we consider two general displacements 
without limitations on the variations. Thus the special form of the 
differential ds^, for the immediate choice of coordinates, becomes 
adx^ -h 2hdx^dx^ -f hdx.^ , 
where x^ and x^, in so far as they occur in a, h, b, are not to be subject to 
variation. 
To find the limiting form of F, we can proceed as follows. As/, g, I, m 
do not occur, and as there is no variation of h through x^ and x ^ , all the 
quantities 
"11 
ir 
“12" 
A2 
"22'" 
22 
3 
j 
4 
5 
3 
4 
3 
4 
vanish in present circumstances. To obtain the limiting forms of A^^-i-A, 
