675] 
ON THE FLEFLECNODAL PLANES OF A SURFACE. 
263 
In what follows, I take for greater simplicity h = 0, (viz. x = 0, y = 0 are here the 
tangents to the two curves of curvature at the point in question), and to avoid 
fractions write 2/, 2g in place of f g respectively; the developed equation of the 
quadric surface is thus 
z — \ (ax 2 + by 2 ) + (ax 2 + by 2 ) (gx + fy). 
I consider the parallel surface, obtained by measuring off on the normal a 
dz dz 
constant length k. If, as usual, p, q denote ^ and ^ respectively, then, in general, 
(X, Y, Z) being the coordinates of the point on the parallel surface, 
z * + V(l +p 2 + q 2 )’ 
X = x ^ 
V(i +p 2 + ? 2 )’ 
y _ kq 
y V(1 + p 2 + q 2 )' 
But in the present case 
p = ax + Sagx 2 + 2 afcy + bgy 2 , 
q = by + afx 2 + 2 bgxy + Sbfy 2 , 
whence 
X — x — k (ax + 3agx 2 + 2afxy + bgy 2 ), 
Y = y — k (by + afx 2 + 2bgxy + 3 bfy 2 ); 
or, putting for convenience, 
X = (l — ka)i;, Y = (l—kb)y, 
then, for a first approximation x = f, y = y ; whence, writing 
P = Zagf 1 + 2af£y + bgy 2 , 
Q = a/F + Zbgtjy + 3 bfy 2 , 
we find 
and thence 
, kP ^ kQ 
x ~i + \-/¡a’ y ,+ l-kb 
■p = «-{t + --rkr„ p ) + p = a i+ P 
1 + ka 
— by + 
1 — ka’ 
Q 
1 — kb' 
Hence 
Z- i (Op + h’) + i~ka iP + vQ + ( “ r + W)+fv) 
+ k -U - \ (a 2 % 2 + b 2 y 2 ) 
a%P byQ | 
or, finally, 
1 — ka 1 — kb) ’ 
Z-k = %{a(l - ka)% 2 +b(l - kb) y 2 } + (op + by 2 ) (g£ + fy),