412 
CALCULATION OF THE MINIMUM N.G.F. OF THE BINARY SEVENTHIC. [696 
unaltered when each power x9 is changed into x u ~Q; and the verification thus obtained 
of the value 
Z 1S = 2 + 6x 2 + a? + 2a; 6 + 2a; 8 + x 10 + 6a; 12 + 2a; 14 
is in fact almost a complete verification of the whole work. Some other verifications, 
which present themselves in the course of the work, will be referred to further on. 
We have, therefore, to calculate the coefficients X 0 , X 1} ...,X 18 ; the function cf>(a, x) 
regarded as a function of a is at once decomposed into simple fractions; viz. we have 
,/ 1 - au 2 
^ ' ’ 1 — ax’. 1 — ax 8 .1 — aaf. 1 — ax. 1 — ax~ L . 1 — ax~ s . 1 — ax~ 5 .1 — ax~ 7 
_ x* * __ 1 
1 — x*. 1 — x 6 .1 — x?. 1 — a; 10 .1 — a; 12 .1 — x u 1 — ax 7 
a? 0 1 
1 — x 2 .1 — x A . 1 — x 6 .1 — x 8 .1 — a; 10 .1 — x 12 1 — aaf 
xf_ 1 
1 — a?. (1 — a?) 1 .1 — x 6 . I — a?. 1 — x w 1 — aa? 
a; 18 1 
1 — a?. (1 — a?) 2 . (1 — x 6 ) 2 .1 — a? 1 — ax 
x w 1 
+ 1 — x 2 . (1 — x*) 2 . (1 — x s ) 2 .1 — a? 1 — ax~ l 
x? 1 
1 — x 2 . (1 — a?) 2 .1 — x 6 .1 — a?. 1 — a; 10 1 — ax~ 3 
1 1 _ 
^ 1 — x 2 .1 — a?. 1 — x 6 .1 — x 8 .1 — x w . 1 — x 12 1 — ax~ 5 
1 
1 — a?. 1 — x s . 1 — a?. 1 — x w . 1 — x 12 .1 — x u 1 — ax~ 7 
In order to obtain the expansion of </> (a, x) in the assumed form of an expansion 
in ascending powers of a, we must of course expand the simple fractions ^&c., 
in ascending powers of a, but it requires a little consideration to see that we must 
also expand the ^'-coefficients of these simple fractions in ascending powers of x. For 
instance, as regards the term independent of a, here developing the several coefficients 
as far as a; 18 , the last five terms give (see post) 
-f x 10 + x v2 + 3a; 14 + 5x 16 + 9a; 18 
— a? — x 6 — 3 a? — 4 x w — 8a; 12 — 11a; 14 — 18a? 16 — 24a? 18 
1 + x 2 + 2a? + Sx 6 +5 a? + lx 10 + 11a; 12 + 14a; 14 + 20a; 16 + 26a; 18 
-x~ 2 .-x 2 - a?-2x 6 -2a?- 4x w - 4a; 12 - 6a; 14 - 7a: 16 -10a; 18 
= _ ar-a +1 0 0 0 0 0 0 0 0 0 
viz. the sum is = 1 — x~ 2 as it should be*. 
* To give the last degree of perfection to the beautiful method of Professor Cayley it would seem 
desirable that a proof should be given of the principle illustrated by the example in the text, and the 
nature of the mischief resulting from its neglect clearly pointed out.—Eds. of the A. J. M.