504] ON THE MECHANICAL DESCRIPTION OF CERTAIN SEXTIC CURVES. 141 
and we have 
A + D = 2 (a + bf + 2d 2 - 2c 2 , 
B + G = 2(a-b) 2 + 2d 2 -2c 2 ; 
whence the relation in question. 
Writing u = i, we have 
{v (G — D) + 4obi)* = - 16a 2 6 2 - (A - B) (G - D) 
= -l(A-B + C-D)\ 
whence 
v(C-D)=±±i{+8ab + A-B+C-D} 
= i(C-D) or -i(A-B); 
A-B 
corresponding to « = - i, the values v = — i and v = i ^ 
And in like manner, corresponding to v = i, we have the values u = i and u = —i 
and to v — — i, the values u = — i and u — i ^^ 
It is easy to show that, if u = i + e, v = i + % are the values consecutive to u = v = i, 
then ae+b£= 0: in fact, substituting the foregoing values in the relation between u, v, 
and writing for 8ab its value, —A—B — C+D, we have 
A+B(-l + 2ie) + (A-B-C+D){-l+i(e + Ç)} + C(-l+2iÇ) + D{l-2i(e + Ç)} = 0, 
which is 
= e(A + B-C-D) + Ç(A-B + C-D) = 0; 
or finally ae + 6£=0. And similarly, if u=-i + e, v = -i+£ are the values consecutive 
to u — v — — i, then we have the same relation ae + b£= 0. 
The points at infinity on the sextic curve are those for which 1 + v? or 1 + v 2 , 
or each of these, is = 0; viz. the values of u, v for the six points are 
— % 
u = i + e, — i — e, i 
A-B .A-B 
V =i + Ç, -i-Ç, - l c -D’ 1 C -D’ 
% 
where, instead of u = v — i and u — v = — 'i, I have written down the consecutive values of 
u, v, and as before e, £ are infinitesimals such that ae + 6^=0. 
Suppose that the coordinates x, y of a point on the sextic curve are 
x = a cos a + c sin a + b cos /3 + d sin /3, 
y = a' cos a + c' sin a + b' cos ¡3 + d' sin /3;