6 
ON STEINERS SURFACE. 
[556 
where the first factor gives the node. Equating to zero the second factor, we have 
|Q.Z + (%q - 1 - xj = X 2 - 1 - ^ - 42' + 4q- lj 
= X=i(l-g)(l + 2}); 
or, finally, 
qZ= 2q +1 + ! ± ? V.l-./Ml + 2g)j X, 
giving two real values for all values of q from q = 1 to q = — , (See the Table 
afterwards referred to.) 
We may recapitulate as follows: 
q > 1, or A > |A; the curve is imaginary, but with three real acnodes, answering 
to the acnodal parts of the nodal lines: 
2=1, or A = £h\ the summit appears as a fourth acnode: 
2 < 1 > i> or A < §h > §h; the curve consists of three acnodes and a trigonoid lying 
within the triangle and having the sides of the triangle for bitangents of imaginary 
contact: 
q = \, or A = fh\ the curve consists of three acnodes and a trigonoid having the 
sides of the triangle for osculating tangents: 
2 < ^ > 0, or A<§h>\h\ the curve consists of three conjugate points and an in 
dented trigonoid having the sides of the triangle for bitangents of real contact: 
2 = 0, or A = \li; curve has the summits of the triangle for cusps: 
2 < 0 > — or A<|/t> \h; curve has three crunodes, or say it is a cis-centric trifolium: 
2 = — or A = ; curve has a triple point, or say it is a centric trifolium: 
2 < — -¿j > — 2 > or A < \h > 0 ; curve has three crunodes, or say it is a trans-centric 
trifolium: 
2 = — or A = 0; curve is a two-fold circle : 
2 < — 2> or ^ < 0; the curve becomes again imaginary, consisting of three acnodes 
answering to the acnodal parts of the nodal lines. 
For the better delineation of the series of curves, I calculated the following Table, 
wherein the first column gives a series of values of A : h; the second the corre- 
sponding values of q, = ^ ^; the third the positions of the point of contact, say 
with the side Z= 0, the value of X : F being calculated from the foregoing formula, 
556] 
and the 
Z : X b 
The Table 
where tb~ 
It 
that X' 
Writing 
The ex] 
Y' + dP 
then th