188 
PROFESSOR M. J. M. HILL OX THE LOCUS OF SINGULAR POINTS 
The solution (64) corresponds to the locus of conic nodes. 
In the case of the solutions (65)-(67) the tangent plane at 77 , £ is 2 = £. 
Hence either of the planes 2 — ± (— 8m 3 /a/3y ) ]/2 touches at any given point on it 
three of the surfaces of the system, viz., those whose parameters are given by the 
equations 
a = £ + 2ma -2/3 / 3 ~ y 3 , b = 77 + 2ma _13 y 3 ~- 3 ; 
a = ^ + 2nt(ua" 2/3 / 3 _l/3 , £> = 77 + 2m&j 2 a -]/3 y6“ 2/3 ; 
a = £ + 2 maru~~ ri / 3 “ 1/3 , 6 = 77 + 2 ma>a ~ 113 / 3 ~ 2/3 
Hence by Art. 9, B (i.), each of the factors 
z±(- 8m 3 /a/3y ) 1/2 
may be expected to occur three times in the discriminant. 
This accounts for the presence of the factors 
{2 ~b (— 8m 3 /a/3y ) 1/2 } 3 . {z — (— 8m 3 /a/3y ) 1/2 } 3 
i.e., 
(z z + 8m 3 /a/3y ) 3 
in the discriminant. 
Art. 11.— To 'prove that if B = 0 be the equation of the Biplanar Node Locus, 
A contains B 3 as a factor in general. 
Let £ 77 , £ be a point on the biplanar node locus. Then equations (11)—(14), (41), 
(42) are satisfied. 
The argument of the preceding article applies so far as A and its first differential 
coefficients are concerned. 
But further the values of 8 2 A/3x 2 , 0 2 A fxdij, given by (56), (57) vanish, in virtue of 
the above mentioned equations, except in the case (to be considered presently) where 
the substitutions x — y = 77 , 2 = £ make 
i.e., 
D 
~J)f W ' 
_Dcq ’ Dij _ 
L> [cq, 5j] 
= 0, 
B [[«].[£]] 
( 68 ). 
From the symmetry of the variables it follows that all the other second differential 
coefficients of A also vanish when x=£,y = r),z=£, (or the same results follow by 
Art. 1, Preliminary Theorem A). 
Hence by Art. 1, Preliminary Theorem B, it follows that A contains B 3 as a factor. 
