268 PROFESSOR M. J. M. HILL OH THE LOCUS OF SINGULAR POINTS 
[a, f]X + O^] Y + |>, £]Z = ol 
[A^]XH-[/3,r ? ]Y + [ y 8 5 C]Z = 0j. 
Hence the values of X, Y, Z which, satisfy (188) and (189) also satisfy (190), which 
was to be proved. 
It may be noticed that the equations (196) are those of the tangent planes to the 
surfaces I)//Da = 0, DfjDfi = 0 at £ y, £. 
Art. 28. — To prove that the Surfaces represented by the three fundamental equations 
intersect in three points on the Biplanar Node Locus, unless the Edge of the Biplanar 
Node always touch the Biplanar Node Locus, and then there are four points of 
intersection. 
(A.) In this case 
[£ f] X* + [y, yj\ Y 2 + K, £] Z 2 + 2 [y, Q YZ + 2 K, a ZX + 2 [£ y] XY 
breaks up into the factors 
tefUte^x + MY + K^z} 
- {R vl ±C(R. v7 - R f] bi. vD) (R f] x + R, ,] y + R, gz}. 
Now, since equations (16), (17), (18), (28), (29) are equivalent to only two 
independent equations, 
[MX+MY+K >17 ]Z and Kax+[^]Y + [^]Z 
are linear functions of 
te«]X + fo,a]Y + K,a]Z and [£ j8] X + [y, fl Y + [£, ffl Z. 
Hence the equations (183)—(185) may be reduced to others in which the degrees of 
the lowest terms in X, Y, Z are 3, 1, 1 respectively. Hence there are three intersec¬ 
tions. 
(B.) If, however, the edge of the binode always touches the binode locus, then by 
(126) it follows that 
K a] x + [y, a] Y + K, a] Z is a multiple of [£ 0] X + [y, 0] Y + [£ /3 J Z. 
In this case 
[f.,]X + [,,,]Y + [d]Z and R, f] X + [£,] Y + [f, QZ 
are not, as in the last case, linear functions of 
R,a]X + [,, a ]Y + [£, a ]Z and R, /3] X + fo, /3] Y + [£, /?] Z, 
