234 THE EEV. GEOEGE SALHOX OX QUATEEXAET CTBICS. 
+ 29824 E^DeB^A+216 E^DC^BA®- 49152 E2DCB*-1280 E^DCBAA^-864 PO 
-9552 E^eBA-162 E^PA^+192 E-^PB^A^+1024E+'^B*-5120E+'^BAA 
-48E^PB^A^+27648 ED^-6912 ED^B^+4608ED^CA- 34816 ED^PB 
+ 128 ED^PA^-1152 ED^CB\A+16272 ED^P-6272 ED+'m+7680ED+'^3^ 
+2952 EDPA-32 ED^PB^A^-1920 EDPB’--216 EDOBA^+1408 EEPB^l 
+ 432 EC®B+108 EC®A^-96 EPB^A+256 EPB^+48 EeB^A^-512 D^O 
-64 D^PA+128 D^PB^+576 D^PB+16D-'PB^A+72 DPBA-128 DPB^ 
-216 DP- 27 PA- 16 PB^A. 
1 next proceed to show how the discriminant of the surface is expressed in terms of 
the fundamental invariants A, B, C, D. And it will throw some light on the investiga- 
tion if I first give the solution of the corresponding problem for a ternary cubic expressed 
in the form 
where x-\-y-\-z-\-u=(}. 
The following is a Table of the principal concomitants of this form : — 
Aeonhold’s invariants : 
S=a5cc^, 
T = + c^cPcd + fZ — 2 abcd{ab + ac + «f7 + ic + cd -\-db). 
The Hessian : 
H = bcdyzu + cdazux + dabuxy + abcxyz. 
It is obvious that the points xy, zu ; xz, yu ; xu, yz are pairs of corresponding points on 
the Hessian. The evectant of S, which Mr. Catlet calls the Pippian, is 
bcd{K — /3)(a — 7 )(a — §)+C(7rt((3 — a)(i3 — yXf^ — + fZa7'(7 — a)( 7 — 
-\-abc (^ — «)(§ — |S)(^ — 7 ), 
from which form it can easily be deduced that the lines joining corresponding points 
on the Hessian touch the curve represented by this evectant, as do also the two lines 
into which the polar conic of any point on the Elessian breaks up. 
The first evectant of T (called by Mr. Cayley the Quip plan) is 
- 5)(« - /3)^ - 2(^V^2a - /3 - 7)(2j3 - 7 - aX 27 - a - /3) ; 
and the second evectant of T, that is to say the Eeciprocant, is 
^c^d%oi-f3f-2^d^bc(ci-(3)Xci-yy 
+ 2«^cf7{(o5— 7)(/3— ^) — (a— ^X 7 — /3)}{(a— ^)( 7 — ^) — (a— /3)(^— 7 )} 
{(a— ^)(^— 7 ) — (a — 7)0 — §)}. 
To return now to the problem of finding the discriminant of this plane cubic. This 
is obtained by eliminating the variables between the three polar conics got by differen- 
tiating the equation with regard to x, y, z respectively ; viz. 
ax^—du^, by^—did, cz^—did. 
