AND LINES IN THE INTERSECTIONS OF A SYSTEM OF SURFACES. 161 
In this case the infinite quantity da } /dx occurs in the term 
, i , 
^ dx \ D./J 2 Dx I)« : ox J 
0 pm Dy; _ / py t y\ /d &1 
^' 3 0x|\Dx 2 Daj 2 \D* Doj/ // Deq 2 ] 
_ D r /D^ ny _ / i>y y\ /Dy x 1 
D«[\Da; 2 Dcq 2 yLLcDflq/ // Dap/ 
/2 _ iteite, d f/py, py _ / i>y y\ /py 1 
D 3 /i UcqlVDic 2 D^ 2 // Dap/ 
Dap 
/• _ o A EA_ I)3 A 
/2 D.D Dyj D.r Doj Dx 2 Dtq 
Dtp 2 
, o /» f P lA Y D 3 /! / D 2 /a V D : y, 
/ p 8 /i \ 8 \D*D«i/ /D 2 /i \ 3 \ D.r Da, / Daf 5 
W/ W/ 
/ 
Now it has been shown in (67)-(69) that 
/(Eh 
/Eh 
V I)rq 2 ° T1U j2 j [Da* 
d / 2 
//EA 
Daf 
does not vanish at points on the locus of binodal lines. 
Hence f 2 
i (Eh i p3 /i 
dx \ D* 2 Dx Dcq dx 
does not vanish. 
Similarly f x ^ 
0 /D 2 /„ D 2 / 2 0a 2 
+ 
dx\Dx 2 1 DxD a 3 dx 
The order of the term 
does not vanish. 
D/ 2 D~/ 1 da v 
Dx Dx Dcq dx 
both vanish, but 
cannot be determined in a perfectly general way, for although Df^/Dx vanishes, yet it 
contains dajdx, dct.Jdx, dct.Jdx, which may be infinite, since a 1? a 2 , a 3 are irrational 
functions of the coordinates. 
These results point to the conclusion that d s A/dx s does not vanish at all points on 
the locus of binodal lines. This is readily proved in particular cases. (See Example 5 
below.) 
Hence, at points on the locus of binodal lines, A = 0, 0A /dx = 0, d~±/dx~ = 0; but 
d 3 A/dx 3 0. 
Hence, if B = 0 be the equation of the locus of binodal lines, when that locus is 
also an envelope, A contains B 3 as a factor. 
mdcccxcii.—A Y 
