410 CALCULATION OF THE MINIMUM N.O.F. OF THE BINARY SEVENTHIC. [696 
+ a 24 (2 — x 2 — 4o? — 6a? — 4a? 4- a? 0 4- 5a; 14 ) 
+ a 25 (x — a? — 2 a? — x 7 + 2 a? 4- 3a; 11 + 5a; 13 ) 
4- a 26 (— 1 — 2a? — 2a; 6 — a? + 4a; 10 — x u ) 
4- a? 7 (2a; + 2a? — x 7 +a? + Sx 11 4- a? 3 ) 
+ a 28 (— x 2 — a? — 3a; 6 — So? — x 12 + 2a; 14 ) 
4- a 29 (a? — a? 4- 3a? 4- a; 11 4- 4a; 13 ) 
4- a 30 (a? — a?— x 6 — x 10 4- 2a; 12 — x u ) 
4- a 31 (— a? — a? 4- 2a; 11 4- a? 3 ) 
4- <x 32 (1 + a? 4- 2a; 10 ) 
4- a 33 (— x — a? — a? — x 7 ) 
+ a 34 (a? 4- a? 4- 2a? 4- x w 4- a; 12 ) 
4- a 35 (—a? — a? 1 — x 13 ) 
4- a 36 . x 14 . 
Denominator (as mentioned before) is 
= 1 — ax .1 — aa?. 1 — aa?. 1 — ax 7 .1 — a 4 .1 — a 6 .1 — a 8 .1 — a 10 .1 — a 12 . 
The method of calculation is as follows: write for a moment 
,X l-*r 2 . 
9 \ a > x ) j _ ax 7 _ i _ aiC 5 < 2 — aa?. 1 — ax. 1 — «a; -1 .1 — ax~ 3 .1 — ax~ 5 .1 — ax~ 7 ’ 
then </> (a, x), developed in ascending powers of a, and rejecting from the result all 
negative powers of x, is 
_ 4 4- aZ x 4-... 4- a 3S Z 36 
1 — ax. 1 — aa?. 1 — aa?. I — ax 7 .1 — a 4 .1 — a 6 .1 — a 8 .1 — a 10 .1 — a 12 ’ 
developed in like manner in ascending powers of <x; for the determination of the Z’s 
up to Z 18 we require only the development of <£ (a, x) up to a 18 ; and, assuming that 
each Z is at most of the degree 14 in x, we require the coefficients of the different 
powers of a in </>(a, x) only up to x 14 . Assuming then that </>(a, x) developed in 
ascending powers of a, up to a 18 , rejecting all negative powers of x, and all positive 
powers greater than x 14 , is 
= X 0 + aX 1 4-... 4- a w X w , 
we have 
y , v , . is y Zq 4- aZ 1 +... 4- a lH Z m 
#lt i-aœ. I - aa?. 1 - aa?. 1 - ax 7 . 1 - a 4 .1-a 6 .1 - a 8 .1 - a 10 .1 - a 12 ’ 
or say 
Z 0 4- aZi 4-... + a 18 Z ls = 1 — a 4 .1 — a s . 1 — a 8 .1 — a 10 .1 — a 12 . 
1 — ax. 1 — aa?. 1 — aa?. 1 — ax 7 .(X 0 + aX x 4-... 4- a 18 ^) ; 
viz. developing here the right-hand side as far as a 18 , but in each term rejecting 
the powers of x above x 14 , the coefficients of the several powers a°, a 1 ,..., a 18 give the