648 
DE. J. CASEY ON CYCLIDES AND SPHEEO-QITAETICS. 
and by comparing coefficients we get 
kP—y'—\ 
kx'=v, kz'—cy'= 0. 
Hence we have the following values for x', y', z', 
, ! X ! C V • 
x—y, y— c _ l ‘, z — 
and these values, substituted in the equation of the given quartic, give after reduction 
cV -b c\c - 1 )> 2 + (c- 1 ) V + 2 c(c - 1 )( A (iv + BA + C c\f) = 0, 
the tangential equation of a conic. 
Cor. By reciprocation we get the following not less interesting theorem : — 
If any tangent T to a curve of the fourth class having three double tangents intersect 
its three double tangents in the three points A, B, C, and if a fourth point D be taken on 
T, such that the anharmonic ratio |ABCD} is given , the locus of 1) is a conic section. 
160. If we take any point P on one of the four nodal lines of 2, then through P can 
be drawn two lines of 2), say L, L/ ; let these meet the other nodal lines of 2i in the two 
triads of points a, a <J, a", b, b\ b " ; then, since the lines of 2) are divided equianharmonically 
by its nodal lines, the two ranges are equal, Paa'a", Pbb'b". Hence the lines are con- 
current, ab, a'b 1 , a"b", the point of concurrence, being the vertex of the tetrahedron oppo- 
site to the plane of the node on which is taken the point P. 
161. If we denote the four nodal conics of % by N, N', N", N w , and if J be the section 
of the sphere U made by the face of the tetrahedron on which N lies, P the point where 
a common tangent PP' of J and N touches N, then the lines of 2j which can be drawn 
through P are coincident ; in fact the section of 2 made by the plane of N consists of the 
conic N repeated twice, and of the four common tangents of J and N, the equations of 
the common tangents being 
x 
(113) 
(see Salmon’s ‘ Geometry of Three Dimensions,’ p. 161). Hence it follows from the 
last article that the common tangent PP' meets each of the remaining nodal conics 
N', N", N'", and that the tangents to 1ST, N", N w , at the points where PP' meets them, 
are complanar and concurrent, the point of concurrence being the opposite vertex of the 
tetrahedron. Hence we easily infer the following theorem : — The three nodal conics 
N', N", N'" pass respectively through the three pairs of opposite intersections of the tetragram 
found by common tangents of J and N. Compare art. 34, 124, and art. 38, ‘ Bicircular 
Quartics.’ 
162. From the theorems of this Chapter may be easily inferred properties of Bicircular 
Quartics ; I give a couple of instances. 
1°. Since the anharmonic ratio is constant of the four planes through any line of A 
to the vertices of the tetrahedron, these planes will cut the sphere U in four circles, which 
