106
ON A CERTAIN SEXTIC TORSE.
[436
Process for the Determination of the Unknown Coefficients.
11. At a point of the cubic curve in the plane w = 0, we have
x : y : z = a (9 + a) 3 : h (6 + ft) 3 : c (9 + 7) 3 ;
and the tangent plane at this point is the osculating plane of the curve; that is, it
is the plane
x y z' 10' _ _
{9 + dj 2+ (9 + ftf + (0 + y) a + (6> + 8) 2 ~
if for a moment (x, y', z, w') are the current coordinates of a point in the tangent
plane. But the equation of the tangent plane as deduced from the equation A = 0 is
, dA
C dx
, dA , d A , dA
y T— + Z -J- + w -j— = 0,
dy dz diu
where in the differential coefficients of A, the coordinates (x, y, z, w) are considered
as having the values
x : y : z : w = a (6 + a) 3 : b (6 4- ft) 3 : c (0 + 7) 3 : 0.
Hence, with these values of (x, y, z, w), we have
dA dA dA dA 1 1 1 1
dx ‘ dy ' dz dw {6 + a) 2 ' (9 + ft) 2 ' (9 + y) 3 * (0 + S) 2î
conditions which determine the values of certain of the coefficients of (*$æ, y, z, w) 2 ,
viz. the six coefficients of the terms independent of w ; and when these are known
the values of the remaining four coefficients are at once obtained by symmetry.
12. To develope this process, disregarding the higher powers of w, we may write
A = © + 3w<3> + xyzw (*]£», y, zf,
where © denotes the terms independent of w, 3zc<3> the known terms which contain
the factor w, and xyzw (* \x, y, z)- the unknown terms which contain this same
factor ; the value of (* ~§x, y, z) 2 being clearly = (*$&’, y, z, 0) 2 .
We have, moreover,
© = (g 2 h 2 x + h 2 f 2 y + f 2 g 2 z) 3 \{a 2 x + b~y + c 2 z) 3 — 27a 2 b 2 c 2 xyz],
= y 4 (b-y + c 2 z ) 2 [a 2 / 2 (Dy 2 — 7h-g-yz + g 4 z 2 )(b 2 y + c-z) + b 2 c 2 (h 2 y + g 2 z) 3 ]
+ g* (a 2 x + c 2 z ) 2 [b‘-g 2 (Dx 2 — 7h 2 f 2 xz + f i z 2 ) (a?x + c 2 z) + c 2 a? (h 2 x + f 2 z) :i ]
4- D (a 2 x + b 2 y) 2 [c 2 h 2 (g 4 x 2 — 7g 2 h 2 xy + f A y 2 ) (a 2 x 4- b 2 y) 4- a?b 2 (g'x + f 2 y) 3 \
— DDg 4 (b 2 g 2 + c 2 h 2 ) x 5
— b 6 Df* (c 2 h 2 4- a 2 / 2 ) y 5
~ (<Pf 2 + % 2 ) * 5 -
