386 ON THE FORTY-EIGHT COORDINATES OF A CUBIC CURVE IN SPACE. 
To obtain two more, write for the moment 
I A y, 8 | =p~\ | y, 8 ! ~q~\ \ A § I =>’ - \ i a , A y \ =s _1 • (23) 
and then multiply (17) by pD x , qD y , ?\D. respectively and add them together. This 
will give 
£tpD.,+ i3pD (/ + (Prlk=0. 
And if we proceed in a similar way with the corresponding systems derived from 
U, Y, W respectively, we may form the following system 
l&k y q-\-(&A z r— 3BA^s=0, iH, 33, . . . suffix x (24) 
£tfby ,_ b . -j-dTB.r— 13 13^=0, ., ,, y, 
^tQ+) + 33C yCj-\- • OC /5 =0, ,, ,, Z, 
^D+>+33D y ^+(!JD ;5 r+ . =0, ,, ,, t, 
whence eliminating p, q, r, s, we finally obtain the following relation, 
33 A y , Qtk z , iOA/=0, £t, 33, . . . suffix x . (25) 
&B,, . ^TB ; , 13B^ ,, „ y 
&C,, 33C y , . 33 (A ,, ,, z 
SID.,., 33D y , (PD.-, „ £ 
To this may be added the corresponding relation obtained from the forms U', V', 
W', T\ These added to the former conditions give us 32 + 2 = 34. 
It was however remarked at the outset that the equations UP' — U'P=0, &c., are 
lineo-linear in the U coordinates, and also in the U' coordinates ; and as we are con¬ 
cerned with the ratios only of the coefficients, and not with their absolute values, we 
are in fact concerned only with the ratios of the U coordinates inter se, and with 
those of the U 'coordinates inter se, and not with their absolute values. Hence the 
number of independent coordinates will finally be reduced to 
48 — 34 — 2=12, 
as it should be. 
