AND LINES IN THE INTERSECTIONS OF A SYSTEM OF SURFACES. 179 
8 ?1 _ 8% 
X - | = Y -v ~ Z - £ 
(27). 
where X, Y, Z are current coordinates. 
Since (25) and (26) are satisfied at all points of the conic node locus, they are 
satisfied when 7 /, £, a, /3 are replaced by £ + S£ rj + 877 , £ -f S£, « + 8 a, /3 + S/3 
respectively. 
Hence 
[a, £] (S£) + [a, 77] (S77) + [a, £] (8£) + [a, a] (8a) + [a, ftftSft = 0 . ( 28 ). 
[ft £] (8£) + [ft 77] (877) + [ft £] (S£) 4 - [ft a] (Sa) 4- [ft ft] (8ft = 0 . ( 29 ). 
Now (28) and (29) are not independent of (16)-(18). 
For if Sa, 8/3 are definite infinitely small quantities, then (16)—(18) determine the 
values of 8 ^, 877 , S£ corresponding to the conic node on the surface ( 10 ). Substituting 
these values in (28) and (29), and observing that Sa, 8/3 being independent may be 
supposed to vanish separately, the following relations are obtained (using the usual 
notation for Jacobians) :— 
^i i| ] ,M,m.H ] ==0 . (30) , 
u L z . v , S - * J 
D[[g.M, [a im = 
D [ £ > v » f, « ] 
WMIMJ-n ' /,o) 
D[ f, V, K, B ]“ u . 
Other similar relations exist which may be found by taking any four of the 
expiations (16), (17), (18), (28), (29), putting any one of the five quantities 8 £, 877 , S£, 
8 a, 8/3 equal to zero, and expressing that the equations give consistent values for the 
four quantities which remain. 
Hence any minor of the fourth order of the Jacobian 
[g,H>[/ 3 ]i 
h[£, 77 , £ , « , /? ] 
vanishes. 
Art. 5 .-—Investigation of the conditions which are satisfied at any point on the Locus 
of Biplanar Nodes. 
The equation of the tangent cone which, in this case, becomes the equation of the 
biplanes, at the singular point is 
2 A 2 
