454]
A THIRD MEMOIR ON QUARTIC SURFACES.
279
where A may be considered as standing for q 2 +gt 3 t 4 . The equation Aw 2 + 2Bw -f T = 0
of the surface, substituting throughout for A its value, is therefore
K\*
+ Jr [t 2 (q 2 + gUi) + 2tpqr] = 0,
(q 2 + gtzk) w 2 + 2 (jpJ {pqr + t (q 2 + gt 3 t 4 )) w
+
K
mm
where the cone is
f / n . + ^ \ , m „, mm' „
a a (q 2 + gt 3 t 4 ) — a -j— r 2 £ 4 — cr ^ p 2 t 3 + j-j-pr
f
fgh J fgh
* (f + gUt 4 ) - jp 2 t 3 | j <*' (q 2 +gt 3 t 4 ) - j- r 2 £ 4 j =
= 0.
152. Writing in the equation of the surface w
K\i .
fgh
instead of w, it becomes
(q 2 + gt 3 t 4 ) w 2 + 2 [pqr + t(q 2 + gt 3 t 4 )] w
+
fg h
mm'
. , . , .. m „, , m „, mm „ ,
0-0- (5- 4- ^ri 3 i 4 ) — a j-rH 4 — a -fP% + p r
f‘
fg h
+ i 2 (g- 2 + #i 3 i 4 ) + = 0 ;
and then writing ^ o- and <7' for o- and a' respectively, this is
(q 2 + gt 3 t 4 ) w 2 + 2pqrw + 2tw (q 2 + gt 3 t 4 )
+ g [<r<r' (q 2 + gt 3 t 4 ) — or 2 t 4 - a'p 2 1 3 + i p 2 r 2 ] + t 2 (q 2 + gt 3 t 4 ) + 2tpqr = 0.
We may consider t 3 , t 4 as denoting not the functions originally so represented, but
these functions each multiplied by a suitable constant, and thereupon write g = — 1 ;
viz., t 3 = 0, t 4 = 0, will now denote any two tangents to the conic A = 0, the implicit
factors being so determined that A = q 2 — t 3 t 4 . The equation of the surface is
viz., this is
(q 2 — t 3 t 4 ) w 2 + 2pqrw + 2tw (q 2 — t 3 t 4 )
— a a (q 2 - t 3 t 4 ) + ar% + a'p% +p 2 r 2 + t 2 (q 2 - t 3 t 4 ) + 2pqrt = 0 ;
(q 2 — t 3 t 4 ) [(w + t) 2 — era'] + 2pqr (w + t) + ar 2 t 4 + a'p 2 t 3 +p 2 r 2 = 0,
the sextic cone being
[a (q 2 - t 3 t 4 ) -p 2 t 3 } [a (q 2 - t 3 t 4 ) - r 2 £ 4 } = 0.
153. But the foregoing equation of the surface is
— a', w + t, . , r
W + t, -a, p ,
• , p , t 4 , -q
t , • > q>
= 0,
as is at once seen by developing the determinant; the functions w + t, a, a', p, q, r, t 3 , t 4
are all of them linear; and the determinant is thus a symmetrical quartic determinant
the terms whereof are linear functions of the coordinates; viz. the surface is a
