148 PROFESSOR M. J. M. HILL OH THE LOCUS OF SINGULAR POINTS 
Art. 4 .— To find the conditions which hold at every point on a Surface Locus of 
Unodal Lines. 
At such a point the tangent cone, whose equation is 
R f] (X - f)« + Ii, ,,] (Y - -nf + R, {] (z - If 
2 Ha(Y-’ 7 )(Z- 0 + 2 B.f](Z- 0 (X-f)+ 2 [f,,](X-f)(Y-,) = D'33), 
breaks up into two coincident planes. 
Hence 
[£ G : K vl ■■ [£ £] 
= bh £] : bh vl : bi> Q 
= K, f| : [C, y] : [l, £].(34). 
Of the four equations (24)-(27), which are satisfied when there is a surface locus of 
binodal lines, it has already been shown that only two are independent. The same 
equations hold when there is a locus of unodal lines. 
Multiply (24) by [ 17 , £], (25) by [£, f], subtract and use (34). Then, 
( 8 a) {[ 77 , f] [£ a] — [£ £] bl> “]} = 0.(35), 
therefore, 
bly £] [£ a ] — B> £] by a ] — 0.(36). 
Similarly 
[U]K«]-[^J[U] = 0.(37). 
By (34) and (36) it follows that (25) depends on (24). 
By (34) and (37) it follows that (26) depends on (24). 
By (34), (36), (37), it follows that, if the values of Sf : S77 : S£ : Sa satisfying 
(24)—(27) are finite, then (27) depends on (24), and the following ratios hold :— 
tt : K v] ■■ K i] : [£ «] 
= by f] : by v] : by G '■ by a ] 
= B. G : B> G ■■ by G : B,«] 
= [«, G ■ [ a , y] ■ l a >G : [a, a ].( 38 ). 
In this case, then, (24)-(27) are equivalent to one independent equation only. 
It may be noticed that in the case in which 
[“. £] = 0, [a, >)] = 0, [a, J] = 0 
(39), 
