AND LINES IN THE INTERSECTIONS OF A SYSTEM OF SURFACES. 
I 57 
The equation (61) is the same as (56). 
Comparing it with (55), which holds when a locus of binodal lines exists, it follows 
that 
{[£ V~] — [£ vf] [«> “] = 0 .(G^- 
Hence [a, a] = 0, or 
[££] [^] - [^? = 0.(63). 
If (63) hold, then by (61) 
[a, ^] [££] = [«, 
therefore 
[£ mi v] = k vViv, vi = [«, s/o. vi 
Making use of these with (24) and (25), it follows that each of these fractions is 
equal to [£, Q/[rj, £]. 
Hence (24) and (25) are equivalent to one equation only. 
But it was shown that (26) and (27) depend on (24) and (25). Hence (25), (26), 
(27) all depend on (24). 
Hence the ratios (38) hold, and therefore there is a locus of unodal lines. But this 
is not the case under consideration, for it is supposed that there is a locus of binodal, 
not unodal, lines. 
Hence (63) is not satisfied, and, therefore, [a, a] = 0. 
(C.) In this case the values of dctjdx, dajdy, given by (47) and (48), are really 
infinite, for D^f 1 JDa 1 Dx does not vanish necessarily, but D'^fJDa^ = 0. 
Consequently the differential coefficients of A require further examination. 
Now 
D 2 /j da, _ / D!/j y /Did 
Dcq Dx dx ' ' \Daj D x) / Dap 
Since D~f l /Da 1 Dx does not necessarily vanish, it must be shown that f 2 
at points on the locus of binodal lines ; i. e ., 
1®% _ 
= 0 
/(x, y , 2 , or 2 ) 
/D 2 / ( x , y, z, ad 
/ Dad 
when x = £ y = y. z = £, the coordinates of any point on the locus of binodal lines. 
Now cq, cq are the roots of 
D f(x, y, z, a) 
Da 
= 0 . 
which become equal when x — y — y, z — £. 
In this case f(x, y, z, a) = 0 is an equation for a, such that three values become 
equal when x = f, y = y, z = £. (They become equal to the same value as oq, cq.) 
