232 
ON THE THEORY OF RECIPROCAL SURFACES. 
[750 
612. Instead of obtaining the second and third equations as above, we may to 
the value of b (— 4n + 6) add twice the value of b (n — 2); and to twice the value of 
c (— 4<n + 6) add three times the value of c (n — 2), thus obtaining equations free from 
p and a respectively; these equations are 
b (— 2n + 2) = 4k — 2b 2 — 5/3 — Si + Qt — j, 
c (— 5n + 6) = 12h — 6c 2 — 5y — 4i — 2 X + SO — 3<w, 
equations which, introducing therein the values of q and r, may also be written 
b (2n — 4) =2q + 5/3 + 67 + Qt + Si +j + 4f, 
c (5n — 12) + SO = 6r + 18/3 + 5y + 4i + 2% + Sw. 
Considering as given, n the order of the surface; the nodal curve, with its singularities 
b, k, f, t; the cuspidal curve, with its singularities c, h; and the quantities /3, 7, i 
which relate to the intersections of the nodal and cuspidal curves; the first of the 
two equations gives j, the number of pinch-points, being singularities of the nodal 
curve, quoad the surface; and the second equation establishes a relation between 
0, a, the numbers of singular points of the cuspidal curve quoad the surface. 
In the case of a nodal curve only, if this be a complete intersection P = 0, Q = 0, 
the equation of the surface is (^1, B, G\P, Q) 2 = 0, and the first equation is 
b (— 211 + 2) = 4k— 2b 2 + 6£ — j; 
or, assuming t= 0, say j= 2 (n — 1)b — 26 2 + 4&, which may be verified; and so in the 
case of a cuspidal curve only, when this is a complete intersection P = 0, Q = 0, the 
equation of the surface is {A, B, G$P, Q) 2 = 0, where AG— B 2 = MB + KQ; and the 
second equation is 
c (— on + 6) = 12 h — 6c 2 — 2% + S0 — 3 w, 
or, say 2% + Sco = (5n — 6) c — 6c 2 + 12h + 30, which may also be verified. 
613. We may in the first instance out of the 46 quantities consider as given 
the 14 quantities 
n : b, k, f t : c, h, 0, % '■ & 7» i ' G, B, 
then of the 26 relations, 17 determine the 17 quantities 
a, 8, k, p, a : j, q : r, w 
n' : a', S', k : b’, f:c' : 1 ; 
and there remain the 9 equations 
(18), (19), (20), (21), (22), (23), (24), (25), (28), 
connecting the 15 quantities 
p', </ : k\ t\ j\ q : hf, 0', X ', / : /3', 7 : V, B\