■HÉ 
[556 
id factor, we have 
lq - l| 
= — i- (See the Table 
eal acnodes, answering 
and a trigonoid lying 
itangents of imaginary 
trigonoid having the 
ate points and an in- 
real contact : 
ps: 
a cis-centric trifolium: 
trie trifolium : 
it is a trans-centric 
ing of three acnodes 
the following Table, 
le second the corre- 
aoint of contact, say 
foregoing formula, 
556] 
ON STEINERS SURFACE. 
and the fourth the apsidal distances, say for the radius vector X = Y, the value of 
Z : X being calculated from the foregoing formula 
The Table is : 
X : h 
z+x 2 + I + -„ + ? v /(i-l)(2 + I). 
q q- q V \q J\ qj 
Contact, 
Z = 0, X : Y= 
imposs. 
imposs. 
1- 
Apses, 
X=Y; X :Z = 
where the asterisks show the critical values of A. : h. 
It is worth while to transform the equation to new coordinates X', Y', Z' such 
that X' = 0, Y' — 0, Z' = 0 represent the sides of the triangle formed by the three nodes. 
Writing for shortness X + Y+ Z = P, YZ+ZX + XY=Q, XYZ—R, the equation is 
(q*P-Qf=4 ! (2q+l)PR. 
The expressions of X, Y, Z in terms of the new coordinates are of the form X' + OP', 
Y' + OP', Z' + OP', where P' = X'+ Y' + Z'; writing also Q' = Y'Z'+Z'X'+X'Y', R' = X'Y'Z', 
then the values of P, Q, R are 
(1+3 0)P', Q' + (20 + 3d 2 ) P'\ R'+OP'Q' + (0*+Ô*)P',