210 PROFESSOR M. J. M. HILL OH THE LOCHS OF SIHGULA.R POIHTS 
Hence dividing out by (X^ — ) 3 , it follows that the left-hand side of equation 
(119) is equal to 
X^ 4- j u, 1 M 1 + zqN4 X l L 3 4* p-4~ tqHh I 
XoL^ 4~ p^M-l 4- ^2-^1 X^Lj "4” d'2^-2 4“ ^£-^2 
Hence if the left-hand side of equation (119) vanish, so also does 
XjLj 4- p-iMj -f- v l N l X l L 3 4" iqN 2 
XoL^ 4“ /u-gM] 4" i X 2 L 2 4" p^Mi 2 ~b 
which was to be shown. 
Hence either method of proceeding will lead to the condition that the tangent 
plane to the conic node locus should touch the tangent cone at the conic node. 
The second method being simpler in this case will now be adopted. 
Multiply ( 1 G) by Si, (17) by Sy, (18) by S£, (28) by — (Sa), (29) by — 8/3, 
and add. 
Hence 
[££] (Sl^ + fo 1 ?]^) 3 + [£>£] (S£)"-f 2 [ 7 ], £] (8r))(8£) + 2[£, i] (S£) Si) 4 - 2 [i, y](Si)(Sy) 
= [a, «] (Sa)" 4- 2 [a, ft] (Sa) (8/3) 4 - [ft, ft] (Sft) 2 . 
Hence the result of substituting the values of Si, Sr 7, SC, which satisfy (16), (17), 
(18), (28), (29) in (120) is 
[a, a] (Sa ) 3 -f 2 [a, ft] (Sa) (Sft) -f [ft, ft] (Sft ) 2 = C. 
Forming the condition that the roots of this quadratic in Sa/Sft should be equal, it 
follows that 
[a, a] [ft. ft] — [a, ft] 2 = (). 
It will be proved in Art. 27 (see the equations (19G)) that the common tangent line 
to the conic node and the conic node locus is in this case given by the equations 
[a, f] (X - i) 4 - [«, vl (Y - V ) + [a, £] (Z - £) = 0 , 
[ft, f] (X - i) 4- [ft, v] (Y - v) + [P> Q (Z - 0 = 0 . 
Hence the common tangent line to the conic node and the conic node locus lies on 
the tangent planes to the surfaces Df/Da. = 0 , I)//Dft = 0 ; and it lies obviously on 
the tangent cone to the surface f = 0 . 
(C.) In this case equations (108) and (109) hold. 
Also differentiating (109) 
