AND LINES IN THE INTERSECTIONS OF A SYSTEM OF SURFACES. 243 
Therefore 
_ I /^/ 0a D/ 05\ 
2 \Da 02 D& 02/ 
= i (G£ + LSx) [G (yi + e y) - H (/3£ + Sexy)] 
+ i (H£ + Ley) [H («£ + 8V) - G (Pi + Sexy)]. 
Hence, dividing numerator and denominator by £, and then putting £ = 0, 
l/jy 0® I 5/ 0J\_ (Gey - H&c) 2 + LSa? (Gy - H/3) + Le 7 (Ha - 0/3) _ 
2 \Da 02 + J)b 02/ «e 2 y 2 - 2(38exy + 7 8V 
Now, in the case n— 2, there is a conic node when x — a,y = b, £ = 0, and then 
G = g, H = h, L = 0. 
Hence 
j /D/ 0a D/ 0&\ (yeZ> — hBa) 2 
2 \Da 02 D& 02/ ae 2 & 2 — 2(38eab + 7 8 2 a 2 ' 
Hence 
D/ 0a D/ 06 
Da 02 D6 02 
does not vanish, 
Art. 20 .—To 'prove that under the conditions stated at the head of this Section, if 
the two Surfaces having Conic Nodes coincide, then they are replaced, by a single 
Surface having a Biplanar or a Uniplanar Node. 
If the condition be expressed that the roots of either parametric quadratic be equal, 
then the roots of the other parametric quadratic must also in general be equal ; for 
treating the parameters as coordinates of points in a plane, this amounts to expressing 
that the straight line (143) touches the conic (161). 
In this case then, the two surfaces having conic nodes coincide, and if a, b be the 
values of the parameters corresponding to them, they may be found by finding the 
points of contact of the straight line (143) Avith the conic (161). 
They are therefore given by the equations 
u x a + WJ) + V,, W x a + v x b + U x V,.a 4- VJ> + w x 
u W V 
Now, since the equation (161) may be replaced by (162) or (163), it follows that 
in the above x may be changed into y or z. Hence 
u x a + W Jo + V x : u y a + Nf> + V y : um + W.b + V. : u : W 
= W x a + vJj -f U„ : W y a + v y b + U y : W z a + vdj + IJ~ : W : v. 
2 I 2 
