216 PROFESSOR M. J. M. HILL ON THE LOCUS OF SINGULAR POINTS 
[a, £] (X — £) + [a. 77] (Y — rj) + [a, £] (Z — £) = 0 , 
[Af](X-f) + [A,](Y-,) + [A £] (Z-O = o. 
(These planes, it may be noted, coincide in this case by (126).) 
Hence che edge lies on the tangent planes to the surfaces D/'/Da = 0, D//D/3 = 0, 
and it is obviously a tangent to the surface f — 0. 
(B.) Conversely, if the edge of the binode always touches the binode locus, then 
the condition (76) holds at every point of the binode locus. 
The equation of the edge is given by any two of the three equations (125). 
Hence, if the edge is a tangent line to the locus of binodes, the equations (125) will 
be satisfied by putting 
X = f+8£ Y = .7 + 877, Z = £ + §£, 
the coordinates of a binode near to 77 , £, which lies on the edge of the first binode 
and infinitely near to it. 
Hence 
K i] m + [£ A (*n) + [6 fl (K) = o,“ 
bh £] ( s 0 + bn, n\ An) + bn, t] At) = 0 , > 
It, f] At) + It, n] An) A A C] (S£) = 0 . - 
(128). 
But equations (16), (17), (18), (28), (29), also hold. 
Hence, by (128) it follows that (16), (17), (18) become 
[£ «] A a ) + [£ fi] (S/3) =0 .(129), 
[ 77 , a] ( 8 a) + [ 77 , fi] (S/S) = 0 .(130), 
It, a] (Sa) + [t, 78 ] ( 878 ) = 0 .., (131) 
Hence 
[£ “]/[£ Z 3 ] = [> 7 > «]/[>?, Z 3 ] = [£, «]/[£, /3].(132). 
Now only two of the five equations (16), (17), (18), (28), (29) are independent 
Suppose that (16), (17) are independent. 
Then, since (28), (29) depend on these, relations exist of the form 
[£ a l — x [£ £] + P [£ v] ’ 
[ 77 , a] = X [f, 77 ] + fi [ 77 , 77 ] , 
[£> a ] = ^ [£ t] + P [£, 1 ?] , 
[a, a] = X [£ a] + g [a, 77 ] , 
[a, /S] t= A [£, /3] + /x [/?, 77 ] 
[£ Z 3 ] = /> [£ f\ + & [i, 77 ] , 
[ 77 , /3] = p [£ 77 ] + cr [ 77 , 77 ] , 
It, fi] — P [£ £] + cr [ 77 , £] , 
[a. yS] = p [£ a] + cr [ 77 , a] , 
[fi &] = p A, fi] + cr [ 77 , /3] . 
