AND LINES IN THE INTERSECTIONS OF A SYSTEM OF SURFACES. 217 
Hence, by (132), 
x [£ £1 + /* H v] _ x [£ v] + p [v> v\ _ x [f ?] + p [y, n _ 
P [£ £] + O' [& /> [ft -v] + O- [>, *?] p [£ ?] + O- [« 7 , f] 
Therefore 
[Xo- — /xp] {[£ [ 77 , 77 ] - [£ T 7 ] 3 } = 0 , 
and 
[Xo- - ftp] {[£ £] [77, {] — [£ 17] [£ £]} = 0. 
Hence, unless Xcr — /xp = 0 , it is necessary to have both 
[£ £] ~ [£ vJ = °> 
and 
k a ^ a - [f, v] 11 q = 0 . 
Hence 
k s/k = it, ^ ^ = k aiv> a 
But if these results hold, the two equations taken to determine the edge of the 
binode would be the same, and would not determine it. Supposing then that those 
two equations have been selected, which are independent, this alternative cannot 
hold, and therefore 
Xcr — /xp = 0. 
Therefore 
[£ «] _ [>7. » ] _ [£ «] _ [«, « ] _ [«, /3] _ x 
[£ /3] “ [ v, f3] ~ [£ /3] - [a, /3] “ [& /3] p 
Hence the equations (126) are satisfied, and in particular 
[a, a] [/3, (3] — [a, (3J = 0. 
Hence, if the edge of the binode always touch the binode locus, the condition (76) 
holds. 
(C.) In this case A is given by (108). 
Let £, 77 , {be any point on the binode locus. 
Then when x = £ y = 77 , 2 = {, 
cq = cq — a of surface having a binode at 77 , {, 
b l — b 2 = b of surface having a binocle at 77 , {. 
J3f/Dx = 0, D/]/D y = 0 , D/^/Ds = 0 , when x — y = rj, z — 
DfJDx = DfJDx, when 2 : = y = 77 , 2 = {. 
Therefore, D/o/Tte = 0 , and similarly D/ 3 /Dy = 0 , D/ 0 /D 2 = 0 , when = £, y = 77 , 
* = C 
2 F 
MDCCCXCII.-A. 
