DE. J. CASEY ON CYCLIDES AND SPHEEO-QUAETICS. 
647 
L, IV, L", U", L iv be five lines of 2, the plane joining ~L to any of the four centres of inver- 
sion will intersect the planes joining 17, L", &c. to the same centre in four lines , whose 
anharmonic ratio will be independent of the centre used in the construction , or, in other 
words, will be the same for all the centres. 
167. Since tlie locus of all the points on two lines of A is a system of four plane 
curves, each of the fourth degree, and having three double points, it follows by recipro- 
cation that the envelope of all the planes through two lines of 2 is a system of four cones , 
each of the fourth class, and each cone having one of the vertices of the tetrahedron for 
vertex, and the three faces which meet in that vertex as double tangent planes. 
158. If L he a line of A, then the anharmonic ratio is constant of the pencil of planes 
through L to the vertices of the tetrahedron. Hence any face of the tetrahedron will 
intersect this pencil in four lines whose anharmonic ratio is constant. Now of the rays 
(four lines), three are lines from the point where L pierces the face of the tetrahedron 
to the three vertices in that face, and the fourth is a tangent to the cone in which the 
same face intersects one of the four cones through WU, — the three vertices forming a self- 
conjugate triangle with respect to that conic. Hence we have the following theorem : — 
If from any point in one of the four nodal lines of A three lines be dr aim to the three 
double points of that nodal line, and a fourth line be drawn tangential to the conic in 
which the fourth cone through WU pierces that face, then is constant the anharmonic 
ratio of the pencil thus formed. 
159. The following direct proof of the converse of the theorem of the last article, 
stated as a property of any quartic curve having three double points, was communicated 
to me by my friend J. C. Malet, Scholar of Trinity College, Dublin. If from any point 
of a trinodal plane quartic three rays of a given anharmonic pencil be drawn to the nodes, 
the envelope of the fourth ray is a conic section. 
Let the quartic be given by the equation 
x^y'-j-y-z' -f- z~x 2J r 2(xyz)(Ax-{-~By -j- C.s), 
where (xy), ( yz ), ( zx ) are the three nodes, and let the point from which the pencil is 
drawn be x 1 y' z! , then three of the rays are evidently the system of determinants 
x J , y } . 
j 
* ? 
X, y, z. 
and these may be denoted by the concurrent systems L=0, M=0, L;7 + My=0. 
Now, if we denote the fourth ray of the pencil by L-j-HVl, the conditions of the 
question give 
v' 
where c is constant : 
c’ ’ 
but 
L + /t'M x(kz 1 —y') J r yx’ — Jczx' 
^Xx-j-yy — yz suppose; 
