1921-22.] Concomitants of Quadratic Ditferential Forms. 195 
Thus the tale of invariants is 
1,( = 0), oC(-O); 
^2 ’ 1^2 ’ 2^2( ~ ’ 
T T • 
J-3 ’ 1^3 ^ 
^4 ? iC 5 2 C y 
I 5 . 1 I 5 , 2 I 5 ; 
^6 > 1^6 ^ 2^6 ’ 3^6 • 
But owing to the definitions of n, 0, v, we have 
n + 0 + v= 0 
always, whether the condition N + O + V = 0 is retained or is used as an 
invariantive condition. Excluding both of these, so that we have twenty 
independent Riemann-Christofiel symbols, together with the ten coefficients 
of the form, we have thirty quantities in the fifteen equations. There will 
accordingly be fifteen independent integrals; these can be taken to be 
the remaining fifteen quantities I2 , • ■ .,316, which constitute the tale of 
differential invariants of the quadratic differential form 
{a, . . .,n^dx\, dx^, dx^^ 
up to the second order of derivation of its coefficients.* 
35. Down to the present, we have secured, as contributions to the 
complete aggregate, the nine quantities 
S,S,; n.n,; A, A,; A , ^ 2 , ; 
r=l s=l 
the five independent line-covariants (there are six, subject to one relation) 
^P), <P), w{V), x{V), y{V), .(P); 
and the fourteen differential invariants other than A, which has already 
been counted. The complete system is to contain thirty members, or, if we 
insist throughout on the persistence of the relation N + O + V — 0, twenty- 
nine in all. One more constituent member is wanted. 
Manifestly the invariantive operator D can be applied to the line- 
covariants in the same way as it was applied to the invariants. Now 
Dw(P) 
is merely A, already retained ; and D%(P) is zero; so no new constituent 
will arise out of u{V). But 
D/;(P) 
* This selection of the independent invariants, so as to make up the aggregate, was 
effected by using the canonical forms of § 32. 
