1G0 
PROFESSOR M. J. M. HILL ON THE LOCUS OF SINGULAR POINTS 
+ [«, I] (S£) + [a, f\ (8y) -f [a, a, Q (8Q + [a, «, «,] (S“) 
+ • • • 
Now 8a' can be shown to be of the order e I/2 , in the same way that 8a was shown 
to be of this order. 
Hence D f(x, y. z, af)[Da? is of the order e 1/2 . 
Hence the same results as those given in (67), (68), (69) follow. 
(E.) Examination of the Differential Coefficients of A. 
Differentiating (52) with regard to x, 
(70). 
Hence the term 
vanishes by (67) at points on the locus of binodal lines. 
Hence d 2 A/dx* = 0 at points on the locus of binodal lines, if it be assumed that 
D fi/Dx is finite. 
This assumption can be made if a v be finite. 
Again, differentiating (70) with regard to x, 
(71). 
