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92 
[398 
ON A CERTAIN SEXTIC DEVELOPABLE 
We have thence AD — BG = 0 and (substituting in the fourth equation) A (X'ri 4- /riy' + v'z') 
+ G (y'z 4- z'x' 4- riy') = 0 ; each of these equations must contain the equation of the 
cone having (x' = 0, y' = 0, z = 0) for its vertex, and passing through the nodal curve. 
The two equations are of the orders 6 and 4 respectively; and as the curve is a 
quartic curve passing through the vertex in question, the equation of the cone is of 
the order 3. I have not effected the reduction of the sextic equation, but for the 
quartic equation, substituting for A, G their values, this is 
— Qriri + ¡Ay + v'z ) [X'ri 2 (y — z) + /riy' 2 (z — ri) + v'z 2 (z — #')] 
4- {y'z' + z'x 4- xy') \X' 2 ri (y — z ) + /ri 2 y' (z — ri) + v'z {x' — y')] 
+ [friri {y' — z) + v'X' (z' — x') + \'/x' (x' — y')} (x + y' + z) — 0, 
which is easily reduced to 
X' 2 x' (— x 2 + y'z' + z’x + x'y ) {y — z) 
+ y! 2 ÿ (— y' 2 + y'z' + z'x' + x'y') {z — x) 
+ v' 2 z (— z' 2 + y'z + z'x + dy') (x' — y') 
+ fiv' \_(ri + y' + /) (y'z' + z'ri — x'y ) -f x'y'z') (y' — z ) 
4- v'\' [(ri 4- y 4- z) (y'z' 4-z'ri 4-x'y') 4- x'y'z'\ (z — x') 
4- \'fi [(ri 4- y' 4- z) (y'z 4- z'ri 4- x'y) 4- x'y'z'] (ri — y') = 0 ; 
and I have found that this is transformable into 
2 [ri V (V) 4- y' V (/a') 4-z' \J (i/)} x [y'z' *J(\')(jx'y' - v'z) z'ri \!(y!)(v'z' - X'ri) x'y' \J(v')(X'x' - y!y) 
- x'y'z' {V OO - V (✓)} {V (ri) - V (ri)} {V (ri) - v (V)}] = 0, 
viz. the two functions are equivalent in virtue of the relation \/ (X') 4- \J (/ri) + ri ( v ') = 0, 
or, what is the same thing, they only differ by a function (ri, y', z') 4 into the 
evanescent factor V 2 + /ri 2 4- v' 2 — 2/riv — 2v'X' — 2X'/ri. The function in { } equated to 
zero is therefore the equation of the cubic cone. 
I do not stop to give the steps of the investigation in the above form, as the 
investigation may be very much simplified as follows: by linear combinations of the 
four equations in ri, y, z', w', 6', we deduce 
( X' — /ri — v) (ri + w' — y' — z’ ) + 26' (y'z' — x'w') = 0, 
(— X' 4~ /ri — v') (y' 4- w' — z' — ri) + 20' (z'ri — y'w) = 0, 
(— X' — /ri 4- ri) (z' + w' — ri — y') 4- 20' (x'y' — z'w') = 0, 
( X' + /ri + ri) (ri + y' + z + w') + 20' (y'z' 4- z'ri + x'y 4- x'w' + y'w' 4- z'w') = 0. 
Hence writing 
X = 
?4_ 
f 
f 
x —w' + ri — y' — z, 
- V + /ri - ri, 
y = w' — ri + y' — z', 
v = 
— X' — /ri + ri, 
z =w' — ri — y + z', 
w — w' + ri +y' + z,