588 ON THE THEORY OF RECIPROCAL SURFACES. [416 
and substituting these values for k and 8, and for a its value =n(n — 1) — 26 — 3c 
we obtain the values of n, c', b'; viz. the value of n is 
n =n ('n — l) 2 — n (76 + 12c) 4- 4b 2 4- 8b + 9c 2 4- 15c 
— 8k— 18h + 18/S 4- 127 + 12i - 91 
-2C-3B-30. 
Observe that the effect of a cnicnode 0 is to reduce the class by 2, and that of a 
binode B to reduce it by 3. 
631. We have 
(n — 2)(n — 3) = 11 2 — n + (— 4n. + 6) = a + 2b + 3c 4- (— 4n 4- 6), 
and making this substitution in the equations (10), (11), (12), which contain (n — 2) (n — 3), 
these become 
a (— 4ra + 6) = 2 (8 — C) — a 2 — 4p — 9cr — 2j — 3y — loco, 
b (— 4<n + 6) = 4& — 2b" — 9/3 — 67 — 3i — 2p —j, 
c (— 4n 4- 6) = 6h — 3c 2 — 6/3 — 4y — 2i — 3cr — ^ — 3«, 
(the foregoing equations (C) Salmon p. 586) ; and adding to each equation four times 
the corresponding equation with the factor (n — 2), these become 
a 2 — 2a = 2 (8 — C) 4- 4 (« — B) — a — 2j — 3^ — 3«, 
2b 2 — 2b = 4& — /3 4- 67 4- 12i — 3» 4- 2p — j, 
3c 2 — 2c = 6/i 4-10/3 4- 4# — 2i 4- 5<r — ^ 4- (o. 
Writing in the first of these a? — 2a = n + 28 + 3/c — a, and reducing the other two by 
means of the values of q, r, the equations become 
n — a — — 2 C — 4 B + re — cr — 2j — 3^ — 3&>, 
2q + ¡3 4- 3i 4-j = 2p, 
3r 4- c 4- + % = 5cr 4- /3 4- 40 4- &>, 
which give at once the last three of the 8 2-equations. 
The reciprocal of the first of these is 
<7' = a — n + k — 2j — 3^' — 20' — 4 B' — 3&/, 
or writing herein a = n (n — 1) — 2b — 3c and k! = 3n (n — 2) — 66 — 8c, this is 
<7' = 4n (n - 2) - 8b - 11c - 2/ - 3 X ' - '20' - 4B' - 3a>', 
giving the order of the spinode curve; viz. for a surface of the order n without 
singularities this is = 4n (n — 2), the product of the orders of the surface and its Hessian.