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A MEMOIR ON THE THEORY OF RECIPROCAL SURFACES.
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9. It is clear that p will in like manner denote the order of the node-couple
curve.
10. I express in terms of
n, 6, c, h, k, /3, 7 , j, 6, x, C, B
such quantities and combinations of quantities as can be so expressed. We have
a = a = n(n— 1) — 2b — 3c,
k = on (n — 2) — 66 — 8c,
S' = \n (n - 2) (n 2 - 9) - (n 2 - n - 6) (26 + 3c) 4- 2b (b - 1) + 66c + fc (c - 1),
4 i = 126 + c (on — 6) — 6c 2 — 07 + 3 6 — 2x>
246 = (- 8n + 8) 6 + (Ion. - 18) c + 86 2 - 18c 2 - 2 (8* - 186) + 20/3 - 15 7 + 4j 4- 96» + 6%,
q = 6 2 — b — 26 — 37 — 6i, (i supra),
r = c 2 — c — 2h — 3/3,
2<r = c (n — 2) — (4/3 + 7) — 0,
8p = (1 Qn- 24) 6 + (- Ion + 18) c - 86 2 + 18c 2 + 2 (8& - 18h) - 9 (4/3 + 7) - 4j - 90 - 6%,
8« = 8n (n - 1) (n - 2) + 6 (- 32m + 56) + c (- I7n + 46) + 86 2 - 18c 2
- 2 (8fc - 186.) + 17 (4/3 + 7) + 4>j + 176 + Q X + 8B >
2S> = n (n — 1) (n — 2) (m — 3) + 6 (— 4/i 2 + 20/i — 24) + c (— 6m 2 + 15m — 18) + 126c + 18c 2
+ {8k - 18h) - 9 (4/3 + 7) - 96 + 2C,
8n = 8m (m - l) 2 + (- 32m + 40) 6 + (- 21m + 30) c + 86 2 - 18c 2
- 2 {8k - 18h) + 21 (4/3 + 7) - 12j + 216-18 X ~ 16(7 - 24B,
c' = 4m (m — 1) {n — 2) + (— 16m + 28) 6 + (— 10m + 26) c + 46 2 — 9c 2
- {8k - 186) + 10 (4/3 + 7) - 4j + 106» - 6 X ~ 6C7 - 8B,
2b' — — a + n (n — 1) — 3c', (m', c' supra),
o-' + 2j' + + 2(7' + 4B' = 4m (m - 2) - 86 - 11c,
p' - 4/ - 6>x - 4(7' — OB' = — 1 1m (m — 2) + a (n' — 2) + 226 + 30c, {n, a supra),
2a-' + 4/3' + 7' + 6'= c' (m' — 2), (m', c' supra),
46' - 3 {%' + 3/3' + 2 7 ') - 2p' - / = (- 4m' + 6) 6' + 2b' 2 , (»', 6' supra),
66-2 (¿' + 3/3' + 27') - 3o"' - x' = (- 4m' + 6) c' + 3c' 2 , (n', c' supra);
{or in place of either of these,
8k' - 186' - 4p' + 9<r' - 2j' + Sx = (26' - 3c') {(m' - 2) (n' - 3) - a}, (m', 6', c', a supra)},
p 4- 2/S' 4- 37' 4- 31' = 6' (m' - 2), (m', 6' supra),
2/^' 4- /3' 4- 3f' +y — 2p =0,
3r' 4- 2i' + %' - 5cr' - /3' - 40' = c', (c' supra),
(twenty-three equations, being a transformation of the original system of twenty-three
equations).
