AND LTNES IN THE INTERSECTIONS OF A SYSTEM OF SURFACES. 
215 
[£ I] (X - i) + [£ r,] (Y - 77 ) + [£ £] (Z - 0 = 0, 
bl, £| (X - 0 + fo, 7?] (Y — 7?) + [ 77 , Q (Z - 0 := 0 , 
B> f] (X - £) + [£, ^ (Y - 77 ) + [c, ( 3 (z-£) = o . . . (125). 
To find the tangent plane to the binode locus, proceed thus :— 
The condition (41) gives 
[£ f] (K a ] [A Z 3 ] — [«> A]' 2 } 
— | £ a ] 2 [(3, /3J + 2 [£ a] [£ /3] [a, /3] — [£ /3]' : [a. a] = 0 , 
which, by means of (76), can be written 
[£ a ] 2 [«, Z 3 ] 2 — 2 [£ a] [£ Z 3 ] [ a > Z 3 ] [a, a ] + [£ Z 3 ] 2 [a, «]’ = 0. 
Therefore 
[£ a] [a, fi] — [£ /3] [a, a] = 0 . 
Similarly 
[ 77 , a] [a, /3] — [ 77 , /3] [a, a] = 0 , 
B» a ] [ a > z 3 ] — B» z 3 ] [«> a ] = °- 
Hence 
VIA _ M [&"] -[«.«] . D8.«] 7 ! 
[£/3] H,/3] [?,/3] [«,/?] [ft/3]. 1 
Now, multiplying (16) by [ 77 , a], (17) by [£ a] and subtracting 
[ 77 , a] {[£ f] (SO + [£ 77 ] (S 77 ) + [£ Q (SC)} 
— [£ a ] [C 7 ?; £] ( S 0 + [>, b] ( §7 ?) + [v> £] (H)} 
+ (SZ 3 ) i[V’ a ] [£ Z 3 ] — bi, Z 3 ] [£ a ]} = °- 
Now, by (126), the coefficient of S/3 vanishes, and the equation of the tangent 
plane to the binode locus is 
R «] SR f] (X - () + R,] (Y - ,) + R 0 (Z - £)} 
-E«]{[,,f](X-f) + [„](Y-,) + [,,i](Z-£)) = 0 . (127). 
Hence the tangent plane to the binode locus passes through the intersection of 
two of the planes (125), and, therefore, through the edge of the binode. 
Hence the edge of the binode always touches the binode locus. 
It may be noticed, further, with respect to the edge of the binode, since equations 
(28) and (29) depend on (LG) and (17), that since it is the intersection of the planes 
(125) it lies also on the planes 
