248 PROFESSOR M. J. M. HILL OH THE LOCHS OF SINGULAR POINTS 
The equation 
U^(uv - W 2 ) + 2 b |>U - VW) + ^ (uw - V 2 ) = 0 
is the only one that need be considered, because the others are identically satisfied. 
In this case 
uv — W~ = — 2xymz — m 2 z 2 
uJJ — VW = myz {2xy + (m — c) z } 
uw — V 3 = ( — m 2 H- 2 me — gf t/ 2 2 2 — 2 mxyh. 
Hence if 77 , 0 be any point on the binode locus, 
0 
^l( uv ~ W 2 ) = — 2m£r), 
~(uU - VW) = 27 n^\ 
^ (uw — V 2 ) = — 2mtjrf. 
Hence the equation for b is 
— 2m^rj (b — 77) 2 = 0 . 
Hence both values of b become equal to 77 . 
Art. 22.— If the two Conic Nodes are replaced by a single Uniplanar Node, and if 
U = 0 be the equation of the Uniplanar Node Locus, then A contains U 4 as a 
factor. 
It follows, as in Art. 19 (B), and Art. 21 , that A and all its differential coefficients of 
the second order vanish. 
Next take the value of 3 3 A/ffr 3 from (150). 
The first three determinants vanish by (158). 
To calculate the next set of terms, put in (177) 
P 'U'zx, q — v xx , r — w xx , P — U xx , Q = V xx , It — W xx . 
Hence they are equal to 
3 (a/-u xx + 2 abW xx + b 2 v xx + 2 aV xx + 2 b\J xx + w xx ) ~ (uv — W 2 ) 
a 
