244 
PROFESSOR M. J. M. HTLL ON THE LOCHS OP SINGULAR POINTS 
Hence the conditions (164) are satisfied. 
Hence there is in general a biplanar node. 
But as a particular case there may be a uniplanar node. 
Art. 21.— If the two Conic Nodes are replaced by a single Biplanar Node, and if 
B — 0 be the equation of the Biplanar Node Locus, and if the Edge of the Biplanar 
Node touch the Biplanar Node Locus, A contains B 3 as a factor. 
It follows as in Art. 19 (B.) that A, dAjdx, 0A /dy, 0A jdz all vanish on the biplanar 
node locus. 
Consider now 0 3 A/0x 3 as given in (148). 
The first three determinants vanish by (158). 
To calculate the next three, put in (177) 
p = u x , q — v x , r — w x , P = U,, Q = V„ R = W, ; . 
Hence these three determinants 
Pi 
= (a 2 u v + 2abW x + b 2 v x + 2 aN x -f- 25IJ, + wf) {uv — W 2 ) = 0 
by (161). 
Next consider 0 2 A/0x dy as given in (149). 
The first three determinants vanish by (158). 
To obtain the next six, put in (177), 
P = Uy, q = Vy, r — Wy, P = Uy, Q = Yy, R = W y. 
Hence their value is 
(cduy + 2abWy + b z v y + 2aV y + 2bJJ y + w y ) (uv — W 2 ) = 0 
by (162). 
Hence by symmetry all the second differential coefficients of A vanish. 
Therefore A contains B 3 as a factor. 
Example 12.-— Locus oj Biplanar Nodes, such that the Edges of the Biplanar Nodes 
always touch the Biplanar Node Locus, the equation of the Surfaces of the System 
being of the Second Degree in the Parameters. 
Let the surfaces be 
(bx — ay + czf — y 2 z 2 — 2 mz (x — a) (y — b) — 0, 
where c. g, m are fixed constants ; a, b are the arbitrary parameters. 
