AND LINES IN THE INTERSECTIONS OF A SYSTEM OF SURFACES. 
165 
Hence, at points on the locus of binorlal lines, i.e., when <£(£ 17 , £) = 0 ; 
, /EA 
/2 / d«j 2 
= /a 
10 8 * 
Art. 9.— Consideration of Loci of Unodal Lines which are also Envelopes. 
(A.) It will be shown that this is the case reserved in Art. 1 , viz., where 
Xd-ffDaf = 0 . 
For if Dyj/Dnp = 0 , i.e., [a, a] — 0 , be substituted in the ratios (38), it follows 
that 
[a, £] = 0, [a, rf -- 0, [a, £] = 0. 
Substituting these in (24)-(26), it follows that 
K £] m + [£ (h) + [£ C] (H) = 0 .(72), 
($£) + bu y] (&n) + lv> £] (§0 = 0 .( 73 ), 
[£, £] m + [£, vl (*v) + K, £] (SO = 0 .(74). 
Now (72)-(74) are equivalent to one equation only by (38). Hence the tangent 
plane to the locus of unodal lines is 
[£ fl (X - O + [£ v] (Y - v ) + [£ 0 (Z - 0 = 0 . 
Now the tangent cone at f rj, £ is given by (53). 
The left-hand side of its equation is by (38) a perfect square. 
Hence the uniplane is 
Kfl(X-f) + [f,,](Y-,) + [a](Z-i)=0 . . . (75). 
Hence the uniplane is the same as the tangent plane to the locus of unodal lines. 
Hence the locus of unodal lines is also an envelope. 
(B.) The converse proposition, viz., that if the locus of unodal lines be also an 
envelope, then [a, a] = 0 , will now be proved. 
If £ + §£, p + Btj, £ + S£ be a point on the locus of unodal lines near to £ 17 , £, 
then the equations (24)-(27) hold. 
If the locus of unodal lines be also an envelope, then the equation of the uniplane 
(75) is satisfied by the values X = £ + Sf Y = 77 + Z = £ + S£. 
Therefore 
[ft f] (Sf) + [ft >;] (8,) + [ft £] (Si) = 0. 
