260 PROFESSOR M J. M. HILL ON THE LOOUS OF SINGULAR POINTS 
to the equation ( 143 ). [The same holds good in the previous article, but the condi¬ 
tion ( 161 ) is not satisfied there.] 
For consider the ratios 
u x a + W J) + V.,: W x a + v.Jo + TJ, = u : W. 
Therefore 
a (W u x - uW x ) + b (WW, - uv x ) + (W V, - uJJ x ) = 0. 
This will be the same equation as ( 143 ) if 
Hence if 
and 
i.e., if 
i.e.y if 
W u x - uW x _ WW X - uv x WV, - uV x 
u ~ W — V 
u r W~ + u 2 v x - 2«WW. f = 0, 
VWW, - Vuv x - W 2 Y, + W7/U, = 0, 
uvu x + u 2 v x — 2uW W, = 0, 
UuW,. — Vuv x — uvV x + WmU, == 0 , 
0 _ 
dx 
(uv — W 2 ) = 0, 
P (TJW - Vv) = 0, 
which are satisfied. 
Similarly the other ratios in ( 164 ) hold. Hence if any point be taken on the curve 
in which the surface (143) intersects the locus of ultimate intersections, that point is 
a binode on the surface (142). Hence the surface (142) has a binodal line situated 
on the locus of ultimate intersections. Hence each surface of the system has a 
binodal line situated on the locus of ultimate intersections. 
It remains to show that if B = 0 be the locus of these binodal lines then A contains 
B 1 as a factor. 
The proof in the last article will hold as far as the second differential coefficients of 
A are concerned. 
Consider, therefore, the value of 0 3 A/3x 3 given in (150). 
The first three terms vanish by ( 158 ). 
The next three are equal to three times 
u„ l (yw - TP) + 2 W.„ t (UV - W w) + | ( uw _ V*) 
+ 2V« |. (W U - Vv) + 2U„ £ (WV - Uu) + w„ V (uv - W 3 ) 
= 0 . 
