185 
1921-22.] Concomitants of Quadratic Differential Forms. 
type must contain twelve members (including the one relation). We shall 
denote the quadratic line complex by u{V), so that 
zi(P) = I)Pd + 2MPiP2+ . . . +2JP5P6 + AP/; 
it is, of course, one of the concomitants of the system. 
When the point-variables are subject to a general linear transformation 
(Xj , X 2 , Xg , X^) = ( , /Xj , , Pi ^ ’ ^2 ’ 
y.j j P2 
P 2,-> ^3 ’ Ps 
^^4 j P'4 ^ 1^4 ? Pi 
Xf, XD, 
where the determinant 0 of the coefficients X, /a, v, p is different from zero, 
the line-variables become subject to the transformation 
fi] 5 • • -5 P0=( (P-2^3)’ (^2^3)’ (hP^s)’ C2P3)’ (/^ 2 P 3 )^ (^Ps) <5 Pfj • • 
i (/^ 3 ^l)’ (^3^1)’ (^3/^1)’ (^sPl)’ (P-sPl)’ (^sPl) 
(p'P'n)’ (^1^2)’ (\p^ 2 )’ (P1P2)’ (V2) 
(p-s^'iX (^sK)’ ihP-d^ (^ 3 Pi)y Q3P4)’ (^3^4) 
(P'2^q), iKP'i), {^ 2 pd’ (p^ 2 Pi)^ i^ 2 pd 
(P'P'i), CD4)’ D1P4)’ (’XP4)’ (P1P4)’ ('^lP 4 ) 
where (/WgVg) denotes M 2 ^ 3 “M 3*^2 5 ^i^d similarly for the other coefficients; 
and then differential operators with respect to the quantities P are subject 
to the transformations 
• • •’ ^ = b PlPi)’ ^1/^4). (>'1^4). (Mi'' 4 ) § P-X- 
' « (Kp,), M, ' 
(X3P4), (/X3P4), (vgpd, (^3^4)’ 0 3^4)’ (P'SPl) ' 
(x^Po), {P1P2), (iqpo), CD2)’ (p-Px) 
(^'^ 3 Pl)’ (psPl)’ (^ 3 Pl)> (^ 3 '^l)’ (p 3 ^p) 
(X2P3), (/X2Ps)> ( 1 ^ 2 P 3 )^ (^^2^3)’ {^' 2 ^)’ (P2^b) 
It thus appears that 
in bracketed pairs, are subject to the same transformations. 
Two useful inferences, among others, can at once be made. The first 
is that the differential operator 
^2 02 02 
sPiSPpePoeppePgSPj 
is covariantive. The second is that, if W and W' be two homogeneous 
line-covariants, the function 
aw aw' aw aw' aw aw' aw aw' ^ 8 W aw aw' 
0p 7 ^ ep; ep3 ePi sp, g^p^ ep^ gpj sp^ sP; 
