The twenty-seven lines upon the cubic surface ... by Archibald Henderson.

52

ANALYTICAL INVESTIGATION

16.25.34 z2 (x1y2w2 - kx2ylw l) x + x1x2z2 (W2- kwl) y
-r 738 +- lY2W2 (+2 - kl( ) Z + XlX22 (Y2 - kyl) W-O
12.34.56 z Z21 (xy2u w2 - kx2y1iv1) x + xxr Zl Z2 (2 - koWl) y
= 739 + 1 xi2 (Y2zl I2 - kyl z21Z ) Z+ - X2Zl Z2 (y2- kl) W -O
14.23.56 y1 Y2 Wl w2 (22 - kzl) x + W1 22 (X2yl1Z2 - k3'lY2z1)
-r4o +Y1Y2 t11 W2 (X2 - kx1) Z+Y lY2 (X2z2w1 - kIXs1z12) w=O
13.25.46 Yl312.t'1V2' (z2- kzl) (xiy2- x2yl) x
7r41 XlZl+Z2W22 (W2 - kClU) (iY'2 - X21) Y
- XIYiY2 W2 (X2 - kxl) (21 tJ2 - Z2 W1) 2
- x1X2Y1Z2 (Y2 - kyl) (Z1 W2- Z2Wl) W=
13.26.45 Y2 Zlwl W2 (2 -- kz1) (X1Y2 - x2Y1) x
7r42 + 2 Z122Wl (1 i2 - kol1) (X1Y2 - X2Y1) Y
- 'V2YiY2il (X2 - kxl) (1 w2 - Z2w1V) 2
- Xl1X2Y2Z1 (Y2 - kl) (ZlW2 - z2w1) 0 =
15.24.36 y, 2Zl '2 (Z2 -kzl) (X1 W'2 - x2W ) X
rr43 + 1x2zl'2 (W2 - kw) (&1Z2 -Y21) Y
+ X2Y1 W1u12 (X2 - kxl) (Yl Z2 -,Y2 Z1) Z
+ -2Y1 1z2 (y2- k1) (X1 2 - x2w1) w= O
16.24.35 yyy2z2wl (z2 - kzl) (xiu2 - x2w1) x
- 44 + xlx2z2wl (%2 - kWl,) (yIZ2-y2z) y
+- X1 Y2 1 2 (X2 - kXl) (Y1 2 -Y2 Zl) Z
+-xly2zz2 (y2 - ky) (x1wi2 - 2wl) w=0
13.24.56 1Y2ZlZ2Vl2 (z2-k1Z) (XlY2 - X2) (x1W-X2W1)
7TX45 +x l;2Z+12z12lI'2 (1 2 -- kWl) (X1./2- X2Y1) (Y12 -Y2Z1) y
-x1 X2YlY21iuW2 (X2 - kex) (YZ12-Y2Z1) (z1lI'2-z2Wl) z
- - 21 /Y2Y2Z1Z2 (Y2 - kyl) (X1V2 - x2u1) (xZ 2- z2'1) w=0
It is clear, from inspection of the equations of the forty-five triple
tangent planes just tabulated, that a perfectly symmetrical system may
be derived by setting k= 1. But the k is retained at this time for a
reason that will appear in the sequel. In fact, in the construction of
the models, the circumstance that k is within our choice enables us,
after assigning numerical values to the other constants, to assign such
a value to k as will bring all the twenty-seven lines within easy reach.
In a word, we use k, so to speak, as a lever.