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696] CALCULATION OF THE MINIMUM N.G.F. OF THE BINARY SEVENTHIC. 409 
where Z 0} Z lt Z 3G are rational and integral functions of x of degrees not exceeding 
14; and where, as will presently be seen, there is a symmetry in regard to the 
terms Z 0 , Z S6 ; Z 1} Z 35 ; &c., equidistant from the middle term Z 1S , such that the 
terms Z 0 ,...,Z 18 being known, the remaining terms Z W ,...,Z 36 can be at once written 
down. 
Using only the foregoing properties, I obtained for the N.G.F. an expression 
which I communicated to Professor Sylvester, and which is published, Comptes Rendus, 
t. lxxxvii. (1878), p. 505, but with an erroneous value for the coefficient of a 7 and 
for that of the corresponding term a 29 .* The correct value is 
Numerator of Minimum N.G.F. is = 
4- a (— x — it 3 — it 5 ) 
+ a 2 {pc 2 4- it 4 + 2it 6 4- it 8 4- it 10 ) 
4- a 3 (— x 7 — a? — x 11 — it 13 ) 
4- a. 4 (2 it 4 + it 8 + it 14 ) 
+ a 5 {x + 2x? — x 9 — if 11 ) 
+ a 6 (— 1 + 2x 2 — it 4 — it 8 — it 10 + x 12 ) 
+ a 7 (4x + a? + 3it 6 — x 9 + x 11 ) 
+ a 8 (2 — x 2 — Sit 8 — Sit 8 — it 10 — it 12 ) 
4- a 9 (it + 3it 3 + it 5 — it 7 + 2it 9 + 2it 13 ) 
+ a 10 (— 1 + 4# 2 — it 6 — 2x? — 2it 10 — it 14 ) 
+ a 11 (5it + 3it 3 + 2it 5 — it 7 — 2it 9 — it 11 4- it 13 ) 
4- tt 12 (5 + it 2 — 4it 6 — 6^ — 4it 10 — it 12 4- 2it 14 ) 
4- tt 13 (it — 4it 5 — 4it 7 — it 9 4- x 11 4- 4it 13 ) 
4- a 14 (2 + 5it 2 4- it 4 4- it 6 — 2it® + 3it 12 — it 14 ) 
+ a 15 (3it — it 3 — x 5 — 7it 7 — 5it 9 — it 11 — it 13 ) 
+ a 16 (6 + 3it 2 4- 3it 4 — 4it 6 — 3it 8 — it 12 + Sir 14 ) 
4- a 17 (—it — 2it 3 — 9it 5 — 8it 7 — 4it 9 — 3it n 4- 4it 13 ) 
+ a 18 (2 4- 6it 2 4- it 4 4- 2it 6 4- 2it® + it 10 4- 6it 12 4- 2it 14 ) 
4- a 19 (4it — 3it 3 — 4it 5 — 8it 7 — 9it 9 — 2it u — it 13 ) 
4- a 20 (5 — it 2 — 3it 6 — 4it 8 4- 3it 10 4- 3it 12 + 6it 14 ) 
4- a 21 (— it — it 3 — 5it® — 7it 7 — it 9 — it 11 + 3it 13 ) 
4- a 22 (— 1 4- 3it 2 — 2it 4 4- x 8 + it 10 4- 5it 12 4- 2it 14 ) 
4- a 23 (4it 4-it 3 — it 5 — 4it 7 — 4it 9 4- it 13 ) 
* The existence of these errors was pointed out to me by Professor Sylvester in a letter dated 13th 
November, 1878. 
C. x. 52