250 
PROFESSOR M. J. M. HILL ON THE LOCUS OF SINGULAR POINTS 
and the next six are 
Hence, B 3 \jBx 3 y Bz = 0 . 
= 2v\u (uv — W~). 
Hence, by symmetry, all the third differential coefficients of A vanish. 
Hence, A contains U 4 as a factor. 
Example 13 .—Locus of Uniplanar Nodes when the equation of the System of Surfaces 
is of the Second Degree in the Parameters. 
Let the surfaces be 
(bx — ay -\- zf — z 3 — 2 mz (x — a) (y — b) = 0 . 
(A.) The Discriminant. 
It is 
r 
! 
a 
1 
s 
(to — 1 ) yz 
- x v 
— mz 
0 
ar 
(to + 1 ) xz 
(to — 
l)yz 
(to + 1 ) xz 
z 2 — z 3 — 2mxyz 
= toz 4 (2 xy + mz - 
- to). 
To show the origin of the factor z 4 , the formation of the discriminant will be 
examined. 
The equations D fJDa = 0 , Df /Db = 0 are, in this case 
ay 2 — b (xy + mz) + (to — 1 ) yz — 0 , 
— a (xy + mz) -f bx 2 + (to + 1 ) xz = 0 . 
Therefore 
a b 1 
— mxz (2 xy + (rn + 1) z} — myz {2 xy + (m — 1 ) 2 } — mz (2 xy + mz) ’ 
Now, it will be shown presently that z — 0 is the uniplanar node locus. Hence, 
a, b become indeterminate on the uniplanar node locus. But, removing the factor 
— mz, which vanishes on this locus, 
2 xy + mz 
Hence, at any point, £ y, 0 on the uniplanar node locus, a = ^, b — y. 
Again, substituting the above values of a, b in 
(bx — ay -f- z) 2 — z 3 — 2 mz (x — a) (y — b), 
a — x 1 -(-7 
2 xy + mzj’ 
b-y 
