156 
PROFESSOR M. J. M. HILL ON THE LOCUS OF SINGULAR POINTS 
K f] (b. £] (X - 0 + [v, v] (Y - v) + fo, Q (Z - 01 
+ { - [£ l\ ± v/ [£ ^]' — [£ £] t 7 ?* ^]} {[£ £] (X — |) + [£ 17] (Y — 77) 
+ [££](z-£)} =0 . (54). 
Now since equation (27) depends on (24) and (25), therefore 
[£f] 
[M 
[£ *] 
bi> f] 
lv> y] 
lv > a ] 
[a, 7 ?] 
[a, a] 
Putting [a, a] = 0 , this becomes 
[“» vJ [£ £] — 2 [ a > [«> £] [£ v] + [«> fp lv> vl = 0 • • • ( 56 )- 
Therefore 
[ a > = {[£ v~\ ± V[€, vJ — [£ £] L 7 ?? v]}l[£> £] • • • ( 57 )- 
Hence the equation of one of the biplanes is 
[«. f] {[£ VI (X - f) + h, >)] (Y - ,) + [£,,] (Z - £)} 
-[M]®fl(X-f) + [G](Y-l) + B£](Z-OS = 0. . (58). 
Now if £ + 8 ^, 77 + 877 , £ + S£ be a point on the locus of binodal lines near to 
££ 77 , £, it follows by (24) and (25) that 
[“. f] i[f. v\ (§f) + Ly, 'll (S'?) + [£, 1 ?] (8£)} 
- U y] {[£ f] m + [f. vl (S'?) + LI £] (8£)} = 0 .(59). 
Hence the tangent plane to the locus of binodal lines takes the same form as (58). 
Hence the tangent plane to the locus of binodal lines is the same as one of the 
biplanes. 
Hence the locus of binodal lines is also an envelope. 
(B.) The converse proposition, viz., that if the locus of binodal lines be also an 
envelope, then \u, a\ = 0 , will now be proved. 
As before, the equations of the biplanes are given by (54), and the tangent plane 
to the locus of binodal lines takes the same form as (58). If, then, the locus of 
binodal lines be also an envelope 
therefore 
{- Li, y] ± Ai v? - [#. f] Ly, I?]} /Li, f] = - [«, ??]/[*, f] • • (60). 
[“. yf [f, f ] — 2 [a, 1 ?] [a, i ] [I, 1 ?] + [a, if [l?, 1 ?] = 0 . . . (61). 
