696] CALCULATION OF THE MINIMUM N.G.F. OF THE BINARY SEVENTHIC. 413 
The expansion is required only as far as x u : the first four terms are therefore 
to be disregarded, and, writing for shortness 
1 
E = 
1 - x 2 . (1 - x i ) 2 (1 - x 6 ) 2 . 1 - a? ’ 
1 
F = 
1 - ¿c 2 . (1 - ar 4 ) 2 .1 - ¿c 6 .1 - a?. 1 - # 10 ’ 
1 
G = 
1 — x 2 .1 — oc*. 1 — x tì . 1 — æ 8 .1 — x 10 .1 — x 12 ’ 
1 
H = 
1 - .1 - Æ) 6 .1 - a?. 1 - x w . 1 - îc 12 .1 - x u ’ 
we have 
n A F G .nr-zH 
which is 
x l0 E (1 + cmt" 1 + a 2 « -2 + ...) 
— a? F (1 + ax~ s + a 2 x~ 6 + ...) 
+ G (1 + ax~ 5 + a?x~ 10 + ...) 
— x~ 2 H (1 -f ax~ 7 + a 2 x~ u +...)> 
where the several series are to be continued up to a 18 , and, after substituting for 
E, F, G, H their expansions in ascending powers of x, we are to reject negative 
powers of x, and also powers higher than x u . The functions E, F, G, H contain 
each of them only even powers of x, and it is easy to see that we require the 
expansions up to x 22 , x u , ¿c 104 and ¿r 142 respectively. For the sake of a verification, I in 
fact calculated E, F up to x M and G, H up to x U2 : viz. we have 
(1 -x 6 )E = (l- x w ) F, 
from the coefficients of E we have those of (1 — x 6 ) E, and in the process of calculating 
F we have at the last step but one the coefficients of (1 — x 10 )F, the agreement 
of the two sets being the verification; similarly, 
(1 - x 2 ) G = (1 - x u ) H 
gives a verification. The process for the calculation of E, 
1 
1 _ x \ (1 _ off (1 _ a*)*. l-x s>