696] CALCULATION OF THE MINIMUM N.G.F. OF THE BINARY SEVENTHIC. 411 
required values Z 0 , Z 18 . We require, therefore, only to know the values of these 
functions X 0 , X lt ..., X w . 
To make a break in the calculation, it is convenient to write 
1 — ax. 1 — as?. 1 — me 5 .1 — ax 7 (X 0 + aX 1 + ... + a l8 X 18 ) = F 0 + aY 1 + ... + a 18 F 18 ; 
putting then 
1 — ax. 1 — aoc 3 . 1 — ax 5 .1 — ax 7 = 1 — ap + o?q — a 3 r, 
where (up to x li ) 
p = x + x? + oc? + x 7 , 
q = x 4 + a? + 2xP 4- x 10 + x v \ 
r = x 9 + x n + x 13 , 
we have 
Y 0 +aY x + a 2 F 2 + ... + a 18 F 18 = (1 — ap -t- a?q - a s r) (X 0 + aX 1 + a 2 X, + ... + a 18 X 18 ). 
The values of F 0 , Y lf . 
f 18 
then are 
F 0 
Fi F 2 
F 3 
f 18 
= x 0 
X 4 X 2 
^3 
- pX 0 - pX x 
-pX. 2 
- 
pX„ 
+ qx 0 
+ qX x 
+ qX i6 
-rX « 
rX 15 
the values being taken 
to x u 
only; and we 1 
bhen have 
Z 0 -f- aZ 1 + a 2 Z 2 + ... + a 18 Zi 8 - 
= 1 - a 4 .1 - a 6 .1 
— a 8 .1 • 
- a 10 .1 - 
<x 12 (F 0 + aF 1 + .. 
viz. the values of Z 0 , Z 1} ..., 
Z 18 are 
z 0 Z x 
Z 2 
¿3 ^4 
Z 7 
Zs 
= F 0 Fj 
f 2 
Y 3 F 4 
f 5 
F 6 
f 7 
Y s 
— F 0 
-Y 1 
-f 2 
— F 3 
-f 4 
— F 0 
- Fj 
-f 2 
— F. 
^3 ^14 
X 16 
Z 17 
o 
II 
f 12 
Fig F 14 
Y» 
F 16 
F 17 
F J8 
— F 6 -F 7 
-f 8 
- F 9 - F 10 
-Y u 
-f J2 
— F 13 
-f 14 
- f 4 - f 5 
— F 6 
- Y 7 - f 8 
-f 9 
— Fj 0 
— F„ 
-f 12 
- f 2 - f 3 
-f 4 
- F 5 - Fi, 
-f 7 
— F s 
- f 9 
— F 10 
+ 214 
+ 2Fj 
+ 2F, 
+ 2F 3 
+ 21 4 
+ 2F„ 
+ 2Fj 
+ 2F 2 
+ F 0 . 
The rule of symmetry, 
before referred 
to, is 
that the 
! coefficient Z 36 _ 
obtained from the coefficient Z p of a p by changing each power xî into x li ~ q , the 
coefficients being unaltered ; in particular Z 1S , the coefficient of a 18 , must remain 
52—2