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PROBLEMS AND SOLUTIONS. 
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form of the curve of intersection of the surface by any other sphere having the same 
centre, and thence the form of the surface itself (being a particular case of Steiner’s 
surface, and which by the homographie transformations w~ x x, w~ l y, w~ l z for sc, y, z 
gives y 2 z 2 + z-x 2 + x 2 y 2 — 2wxyz = 0, the general equation of Steiner’s surface). 
Solution by the Proposer. 
Take X, X', Y, Y’, Z, Z' the intersections of the sphere x 2 + y 2 + z 2 = 1 by the 
three axes respectively; then we have x 2 + y 2 + z 2 = 1, x + y+z — — 1, the equations of 
the circle through the points X', Y', Z'; and from these two equations we deduce 
yz + zx-\-xy = 0, and thence 
that is 
y-z 1 + z-x 1 4- x 2 y 2 + Ixyz (x + y + z) = 0, 
y-z 1 + z 1 x <1 + x-y- — 2 xyz = 0 ; 
so that the circle lies on the quartic surface; and by changing successively the signs 
of each two of the three coordinates, we have three other circles lying on the sphere 
and also on the quartic surface; viz. we have in all four circles; the above-mentioned 
circle through (X', Y', Z'), and three other circles through (XY, Z), (X, Y', Z), 
(X, Y, Z') respectively, making together a curve of the order 8, the complete inter 
section of the quartic surface by the sphere. 
The quartic surface lies entirely in the four octants of space xyz, xy'z', xyz', x'y'z; 
and as to the portion of the surface which lies in the octant xyz, this meets the 
sphere x 2 + y 2 + z 2 = 1 in portions of the three circles (X\ Y, Z) {X, Y', Z) (X, Y, Z') 
constituting a tricuspidal form lying within the octant XYZ as shown in the figure. 
The intersection by a sphere, radius < 1, projected on the octant XYZ, is a trinodal 
form, lying outside the tricuspidal one, as shown by a dotted line in the figure; the 
intersection by a sphere radius > 1, projected in the same way, is a trigonoid form 
lying inside the tricuspidal one, as also shown by a dotted line in the figure; as the 
radius approaches to and ultimately becomes 
A 
V3 ’ 
this diminishes, and becomes 
ultimately a mere point, and Avhen the radius is greater than this value the intersection 
is imaginary. 
Imagine on the solid sphere, radius = 1, the four tricuspidal forms lying in 
alternate octants as above; cut away down to the centre the portions lying without