NOTES AND REFERENCES. 
595 
a p : b q is finite; and this being so the fractional or it may be integer number p : q 
is the fourth parameter in question. But it is preferable to adopt Halphen’s purely 
descriptive definition, viz. we consider a conic 1° in reference to three given points 
y, z, t on a given line, and take x, x for the intersections of the conic with the 
line : we take a = (y, z, t, x) — (y, t, x') for the difference of the corresponding anhar- 
monic ratios of the three points with the points x, x' respectively; and 2° we consider 
the conic in reference to three given lines F, Z, T through a given point and take X, X' 
for the tangents from the given point to the conic; we take 6=(F, Z, T, X)—(F, Z. T, X') 
for the difference of the corresponding anharmonic ratios of the three lines with the 
lines X, X' respectively (observe that these values are a = —°° - X -— ^ - — -, and 
1 J \ z — x.z—x z — y.z — t 
b = 
X-X' 
Y-T 
Here when the conic is a line-pair-point, x = x' and 
F-X.Z-X' ‘ Z — Y.Z-Tj 
X = X', where a = 0 and 6 = 0, but we have as before the integers p and q such that 
a? : b q is finite, and we have thus the fourth parameter p : q. 
Halphen’s correction is now as follows, starting from the formula number of conics 
(X, 4>Z) = ay, + ¡3v, we may have among the + conics line-pair-points any one of 
which if we disregard altogether the fourth parameter is a conic satisfying the five 
conditions, but which unless the fourth parameter thereof has its proper value is an 
improper solution of the problem and as such it has to be rejected: if the number 
of such solutions is = T, then there is this number to be subtracted, and the formula 
becomes, Number of conics (X, 4Z) = ay, + /3v — B 
It may be asked in what way the fourth parameter comes into the question at 
all: as an illustration suppose that a, b denoting the semiaxes of a conic, or else the 
above mentioned descriptively defined quantities, then p, q, k denoting given quantities 
(p and q positive integers prime to each other) the condition X may be that the 
conic shall be such that a p + b q = k\ this implies a p : b q finite, and hence clearly if the 
system of conics (X, 4Z) contains line-pair-points, no such line-pair-point can be a 
proper solution unless this relation a p + b q — k is satisfied. 
412. Zeuthen’s Memoir of 1876 presently referred to contains applications to the 
theory of Cubic Surfaces, the numerical results given in the table p. 539 agree for 
the most part with those of the Memoir 412, see p. 363, but for the surfaces III, VI, IX 
and XII discrepancies occur in the values of r and h' relating to the spinode develope. 
As to this observe that Zeuthen’s li, or say li includes actual as well as apparent 
double planes, and we have r' = c /2 — c — 2li — S/3', my li relates to apparent double 
planes only, but as I assume that there are no actual double planes the formula is 
r' = c' 2 — c— 26' — 3/3', and as the values of c' and ¡3' agree we have in fact in each of 
the four cases r + 2li (Cayley) = r + 2h' (Zeuthen). The values found are 
III 
VI 
IX 
XII 
III 
VI 
IX 
XII 
Cayley n' 
72 
24 
12 
6 
r’ 
42 
24 
32 
' 9 
Zeuthen n 
84 
30 
24 
7 
r 
18 
12 
8 
7