190 Proceedings of the Poyal Society of Edinburgh. [Sess. 
down, by noticing the symmetry in D, A, N, , Pg ; E, B, O, Pg , P 5 ; 
F,C,V,P 3 ,PJ, 
r=.N(DPi 2 +APg 2 ) + (N 2 + AD)PiPg+ . . . , 
24; = (3N2 + AD)(I)Pi 2 + APg 2 ) + 2N(3AD + ,N 2 )P^Pg+ . . . , 
x = 2 N(N 2 + AD)(DPi 2 + APg 2 ) + (N4+61s2AD + A 2 D 2 )P^Pg+ . . . , 
(A‘^D 2 + 10ADN2 + 5Nb(BPp+ APg 2 ) , 
+ 2N(5A2D2 + ioaDX2.^^4)p^P^_,_ _ 
2 = N(3 A2D2 + 10ADN2 + 3 + APg^) 
+ (A3D3+15A2D2,N2+15ADN4 + N«)l\Pg+ . . ., 
Ii=:N + 0 + Ab 
1 2 = N 2 + AD+ . . 
1 3 = N(N 2 + 3AD)+ . . 
I^=N4+6ADN2 + A 2 D 2 + ^ ^ 
l5==N3+10ADN3 + 5A2D3N+ . . 
Ig = N6+15ADN4+15A2D2N2 + A3D3+ . . . 
It is manifest that the six invariants Ij , I 2 , I 3 514 , 15.16 independent 
of one another ; they are the sums (save as to a numerical factor 2) of 
the first, second, third, fourth, fifth, and sixth powers of the six inde- 
pendent quantities 
, h = N ± (AD)’ ; /'3 , /.q = 0 ± (BE)^ ; \ = (CE)t 
(For the form under consideration, the discriminant □ becomes 
□ = (AD - N2)(BE - 02)(CF - V^) ; 
and it is easy to verify that 
□ =ii6 + ih^-W4-tV. 
when we take 1^ = 0.) 
Again, it is not difficult to see that there is no relation among the 
line-covariants, which are linear functions of 
a-^=DPi2pAPg2, cT3 = EP,2 + BP 52 , CT3=FP32 + CP;2, 
0-2 = PiPe 5 — 1*2^5 ’ ^6 ~ ^3^4 ’ 
in the forms 
^ = CTj -H (/l^2 + /i’2)(T2 + . . 
2f^ = /h2)(Tj^ -j- ~h 
2.r=(7rq + A:2)(V + /'^2>i+(V + ^^2V2+ ■ • 
y = (Aq4 -f k^k .2 + k^k^ -1- k-J^\^ -f + (Aj ^ + k, 2 )(T 2 + . . ., 
22=(A,-l-A2)(A,4 + AyA22-l-A2bo-, + (Ai« + A2V2+ • • • 
