DR. J. CASEY ON CYCLIDES AND SPHERO-QTJARTICS. 
645 
If we reciprocate the Cor. in the last article we get the following theorem : — 
If the four centres of inversion of WU he joined by planes to two non-consecutive lines 
of WU, the four lines of intersection of the homologous pairs of planes are generators 
of a ruled quadric. 
149. The lines of divide its nodal conics homographically. 
Demonstration. Let five lines of 5), namely L, L', &c., meet its nodal conics in four 
systems of five points, namely l, l', &c., m, m', See., n, n', See., p,p', Sec . ; then, by art. 147, 
the four systems of lines 
IV , 
mm' , 
nn! , 
PP' > 
ll" , 
min" , 
nn" , 
PP" , 
IV" , 
mm'" , 
nn'" , 
pp'" , 
IV", 
mm"", 
mV", 
pp"" 
are generators of four hyperboloids, H, XL, H", XL", and the planes L lit , HI", L It", Lll"" 
are tangent planes to H, H', &c. ; hence the anharmonic ratios are equal, l {l’, l!', I l""} 
{ H H' II" H'" } . Hence the proposition is proved. 
150. By reciprocating art. 149 we get the following theorem : — If four tangent planes 
he drawn through any four lines of the system WU to one of the four cones through WU, 
the anharmonic ratio of these four tangent planes is equal to the anharmonic ratio of the 
four tangential planes drawn through the same lines of WU to any of the three remaining 
cones. 
Cor. From this proposition we infer the following theorem : — The anharmonic ratio of 
the four edges of one of the four cones of WU passing through any four points on WU 
is equal to the anharmonic ratio of the four edges passing through the same points of any 
of the three remaining cones. 
151. Since the sphero-quartic WU is a curve of taction on "S, the tangent line to WU 
at the point where L cuts it (see art. 146) is a tangent line to the conic of art. 146. This 
theorem maybe enunciated as follows: — A tangent plane to the sphere U at any point P 
of the sphero-quartic WU” intersects the four faces of the tetrahedron whose vertices are 
the centres of inversion of the sphero-quartic in four lines; and the conic determined by 
these lines and the tangent line to WU at P will also touch the line of contact of the 
plane with S at the cuspidal edge of %. 
We shall have to refer so frequently to the tetrahedron formed by the centres 
of inversion of W r U, that in order to avoid circumlocution I shall simply call it the 
tetrahedron. 
152. By reciprocating with respect to U we get from the theorem of the last article 
this other theorem : — The four lines drawn from any point P of a sphero-quartic W r U to 
its four centres of inversion, the line of "S passing through P, and the line of A through 
P are edges of the same cone of the second degree, which has also the osculating plane of 
WU at P for a tangent plane. 
The cone possesses this property also ; namely, the anharmonic ratio is constant of the 
four edges passing through the centres of inversion. 
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