AND LINES IN THE INTERSECTIONS OE A SYSTEM OF SURFACES. 207 
Art. 14. — To prove that if the Conic Node Locus be also an Envelope, A contains C 3 
as a factor. 
(A.) It will be shown that when the condition (76) holds, in the case of a conic 
node locus, then the conic node locus is also an envelope. 
In this case, from (28) and (29), by means of (76), it follows that the equation of 
the tangent plane to the conic node locus is 
[A a] {[a, £] (X - |) + [a, r,] (Y - V ) + [a, £] (Z - Q} 
- [a, a] m f] (X - £) + C/3, 17 ] (Y - 17 ) + [A £] (Z - 0} = 0 (113). 
This will touch the tangent cone (33) at the conic node, if 
[££l [£* 7 ] [££] 
bi>i] tv,v\ bh£] 
Hi] Hv\ [U] 
j[A«][«,f| ] j[A«]M] ] f[A«][«>£] ] 
— [a, a] [/S, J | —[a,a] [/3, 77 ] J 1 — [a, a] [/3, £]j 
This can be written 
[A a ] [a, f] ~ [ a , a ] [A £] 
1 / 3 , a ] [a, 17] —[a, a] [A 17] 
[A a ] [ a , £] — [a, a ] [A Q 
(114). 
Hi] 
[£^ 7 ] 
bi, f] 
[>?> 
K, i] 
K. ^ 
f[A «] [«, i] 1 
J L 
([A a ] [*> 
“ [a, «] [A f]j 
1 - [a, «] [A 17 ]' 
[£fl 
[£*7] 
tv, fl 
bh y] 
[£ fl 
[£ ^ 
I [A «] [«, f] 1 j[A a ][ a >’?] j 
I ~ [*, «] [A f]l | - 0 , a] [A 17]! 
[££] 
[*> a 
tv, ti 
0> v] 
[£ £] 
[*» £] 
|[A a ][«A] ) 
j [A «] [«, a] 
1 — [a, a ] [A £]j 
► 5 
1 — [“, a] [A a . 
[££] 
[a a 
h, a 
[A t?] 
K. £] 
[A £] 
j[A «][«, £] j j[A «] [«» Z 3 ] 
— [a, a] [£, £] 1 1 — [a, a] [£, /3]j 
= 0 
(115). 
For the constituent in the fourth row and column of the last determinant vanishes 
by (76); and the constituent in the fourth row and column of the preceding 
determinant is identically zero. 
Hence the condition becomes 
