521] 
AND ON THE CONSTRUCTION OF A DOUBLE-SIXER, 
383 
or 
abc (x — a) (Fx -I- Hf + (x — b) (x — c) (Gx + Ha) 2 = 0. 
Developing and throwing out the factor x, this is 
G' 2 x 3 
+ {2<z GH — (b + c) G 2 + abc F 2 \ x 2 
+ [a 2 H 2 —2a(b + c) GH + be G 2 + abc (2FH — aF 2 )) x 
+ {— (b + c) a 2 H 2 + 2abc GH + abc (H 2 — 2aFH)} = 0. 
This must be satisfied by x=m, x=m 1 \ hence the left hand must be =G 2 (x—m)(x—m 1 )(x—(r), 
or equating the constant terms we have 
G 2 7)1171! <r = aH {— 2abc F + 2be G + (bc-ab — ac) H), 
which gives a; and we then have 
V2 = - 
cr — a 
G<r + Ha 
{Fa + H), 
but I have not attempted the further reduction of these expressions. 
The numerical values for the example are 
— 140 + 62 Vl4 „ — 10 + 62 Vl4 TT — 104 Vl4 
or = “ 7=^ , Or = ■; : _ 1= , 11 = 
5 + 21 Vl4 ’ 5 + 2lVl4 
whence a as in the Table. 
Point 2' by means of the conic 1362'4'5 / . 
The equation of the conic is 
where 
Fx + Gy + H = - 
— G + H — — be, 
(x — b) {x — c) 
y 
Fm + G + H = ———Q 
Vilf 
Fm 1 -G^/W 1 + H= 1 -~. 6 ) fa ~ c > 
— vMj, 
0 + Vl4 
(4 / , 5') 
(3) 
(1) 
(6), 
which are the same as for point 3\ if only we reverse the signs of F, H and Vjf, V.