[485 
the same 
Steiner’s 
x, y, z 
1 by the 
uations of 
re deduce 
the signs 
he sphere 
mentioned 
f, Y', Z), 
ete inter- 
yz, xyz; 
neets the 
X, Y, Z') 
he figure, 
a, trinodal 
gure; the 
loid form 
e; as the 
becomes 
itersection 
lying in 
r without 
485] 
PROBLEMS AND SOLUTIONS. 
591 
these tricuspidal forms; and build up on the tricuspidal forms, until the greatest 
2 
distance from the centre becomes = — ; we have a solid figure with four prominences 
V O 
situate as the summits of a tetrahedron, the bounding surface whereof is the surface 
in question: it is to be added that the axes are nodal lines on the surface, viz. the 
portions which lie within the solid figure are the intersections of two real sheets of 
the surface, the portions which lie without the solid figure are isolated, or acnodal, lines 
on the surface. 
[Yol. ix. pp. 73, 74.] 
2573. (Proposed by Professor Cayley.)—The envelope of a variable circle having 
for its diameter the double ordinate of a rectangular cubic is a Cartesian. 
[Definition. The expression “a rectangular cubic” is used to express a cubic with 
three real asymptotes, having a diameter at right angles to one of the asymptotes and 
at an angle of 45° to each of the other two asymptotes, viz. the equation of such a 
cubic is xy- = a? + bx 2 + cx + d.} 
Solution by the Proposer. 
The equation of the variable circle may be taken to be 
2A 
(x — 6) 2 + y 2 = 6 2 — 2 md + a + 
6 ’ 
viz. 6 being the abscissa of the rectangular cubic, the squared ordinate is taken to 
be = ^(0 3 — 2m6 2 + u9 + 2A), or, what is the same thing, the equation of the variable 
circle is 
2 A 
x 2 + y 2 — a — 2(x — m) 6 —= 0. 
u 
Hence, taking the derived equation in regard to 6, we have 
x — m — ^ = 0, 
V 
and thence 
therefore 
x 2 + y 2 — a 
4A. 
6 ’ 
16^1 2 
{x 2 + y 2 — a) 2 = = 16 A (x — m); 
that is, the equation of the envelope is 
(x 2 + y 2 — ol) 2 — 16A (x — to) = 0, 
which is a known form of the equation of a Cartesian.