208 
PROFESSOR M. J. M. HILL ON THE LOCUS OF SINGULAR POINTS 
r/ 3 - a ]' 2 
hi tn. [«]} 
— 2 [/3, a] [a, a] 
B {?> v> C, a ) 
hi [a w} 
B {£ v, c, 
, r T vm[ V ],[nm ft 
+ [a> “ ] “IUUU3” - 0 
( 116 ). 
And this is satisiied, because (30)—(32) hold. 
Hence if (76) hold, the tangent plane to the conic node locus touches the tangent 
cone at the conic node. 
Therefore the conic node locus is also an envelope. 
(B.) Conversely, if the tangent plane to the conic node locus always touch the 
tangent cone at the conic node, then the condition 
[a, a] [/3, /3] — [a,- /3J = 0 
is satisfied at every point of the conic node locus. 
To determine the position of the tangent plane to the conic node locus, it is 
sufficient to eliminate Sa, S/3 from any three of the equations (16), (17), (18), (28), 
(29), and then to use the relations (27). 
Suppose that the values of S£, Sr/, SC, which satisfy (16), (17), (18), (28), (29), are 
S£ = X l Sa -f X 2 S/3 
Sr/ = Sa -f p -2 S/3 > 
SC = r, Sa tq S/3 
Then the tangent plane to the conic node locus is 
(117). 
(X — f) (/X, + (Y — t]) (i^X, — 1-oX,) + (Z — Q — Xj/nJ = 0 . (118). 
The condition that this may touch the tangent cone (33) is 
[£ f] 
5?» £] 
K. f] 
Mffis — W 
[£ y\ 
lv> y] 
[££] 
h. G 
[C, a 
^C< u l 
[X^Vo - 
0 
It will now be proved that the same condition can be obtained by substituting the 
values of (S£), (§> 7 ), (SQ from (117) in 
