646 
DR. J. CASEY ON CYCLIDES AND SPHERO-QUAETICS. 
153. Let us now consider the developable A reciprocal of This is formed by the 
tangent lines of WU. Let Iv be one of Eig- 4. 
the centres of inversion of WU, P, P', 
Q, Q! two pairs of inverse points of 
WU ; then, if P, Q be consecutive points, K 
PQ, P'Q' are two lines of A, and their 
point of intersection, S, is a point on two 
lines, and the locus of S will he a double or nodal line of A. 
Now we have seen that, W being aa 2 + bf3 2 + cy 2 -j-dh 2 , U 2 =« 2 -f /3 2 -f-y 2 -f-c5 2 , the sphero- 
quartic WU will be the intersection of the quadrics 
ax 2 + by 2 -j- cz 2 + dw 2 = 0 , 
x 2 -\-. y 2J r z 2J r w 2 = 0, 
and the equation of the nodal line of the developable A is (see Salmon’s 4 Geometry of 
Three Dimensions,’ art. 209) 
(c-c ) 2 (c-a ) 2 (a-b ) 2 
hex 2 cay 2 abz 2 
The same equation may be easily inferred from 4 Bicircular Quartics,’ art. 43. 
Hence each of the four nodal lines of A is a quartic curve having three double points , 
the double points being at the centres of inversion, which are in the plane of the nodal line 
and passing through the four single foci of the sphero-quartic which lie in the plane of 
the nodal line. 
154. Every line of A has a corresponding line in A ; and by art. 146 any line of A and 
the corresponding one of A are tangents to a conic, which also touches the four lines in 
which their plane intersects the faces of the tetrahedron. Hence any line of A, and the 
corresponding line of 2, are divided homographically by the faces of the tetrahedron ; 
but the lines of 2 are divided in a given anharmonic ratio by these faces (see art. 145). 
Hence the lines of A are divided in a given anharmonic ratio by its four nodal lines. 
Cor. If two lines L, L' of A meet its nodal lines in two systems of four points l, m, n, p, 
l', m', n',p', the corresponding chords of the nodal lines ll' , mm', nn',pp' are generators 
of an hyperboloid ; for L, L' are divided equianharmonically by the nodal lines of A. 
155. The nodal or double lines of A are homographic figures. 
For let five lines of A meet its double lines in the four systems of five points each, 
then are equal the four pencils 
l {It l" l '!" l"" }, m{mt m" m 1 " m""}, 
n{n' n" n'" n""}, p { p' p" p'" p"" } ; 
that is, the four pencils are equal which are formed by the corresponding chords of the 
four double lines. This follows exactly in the same way as the corresponding proposition 
of art. 149. Hence the proposition is proved. 
156. By reciprocation we get from the Cor. of art. 154 the following theorem: — If 
