ON THE THEORY OF RECIPROCAL SURFACES. 
589 
416] 
632. 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 (— 4n + 6) add three times the value of c (n — 2), thus obtaining equations free from 
p and cr respectively; these equations are 
b(— 2n + 2) = 4& — 2b 2 — 5/3 — Si + 6t — j, 
c (— 5n + 6) = 12h — 6c 2 — 5y — 4i — 2% 4- 30 — 3co, 
equations which, introducing therein the values of q and r, may also be written 
b (2n — 4) =2q+ 5/3 + 67 + 6£ + 3i + j 4- 4/, 
c (5 n —12) + 30 = 6r + 18/3 + 57 + 4 i + 2% + 3 eo. 
Considering as given, n the order of the surface ; the nodal curve with its singularities 
b, k, f t; the cuspidal curve and 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 6, <w, 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 (A, B, G\P, Q) 2 = 0, and the first equation is 
b (— 2 n + 2) = 4 k — 2 b 2 + 6t —j; 
or, assuming t = 0, say j = 2 (n — 1) b — 2b 2 + 4>k, 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, Qf = 0, where AG — B-= MP + NQ ; and the 
second equation is 
c (— on + 6) = 12h — 6c 2 — 2^ + 30 — 3w y 
or, say 
2^ + 3(o = (5/i — 6) c — 6c 2 + 12/i + 30, 
which may also be verified. 
633. We may in the first instance out of the 46 quantities consider as given 
the 14 quantities 
n ; b, k, /, t ; c, h, 0, x 5 & 7. i 5 G, B, 
then of the 26 relations, 17 determine the 17 quantities 
a, 8, k , p , a ; j, q ; r, w ; 
S', ; o' ; » ¥ , 
and there remain the 9 equations 
(18), (19), (20), (21), (22), (23), (24), (25), (28), 
connecting the 15 quantities 
p, <t' ; k'y t', /, q ; h', 0\ x> r ' 5 P, 1 \ G', B'.