ME. A. CAYLEY’S MEMOIE ON CUEVES OE THE THIED OEDEE. 425 
16. To find the coordinates of the point F, which is the harmonic of G with respect 
to the points E, F. 
The coordinates of the point in question are 
uX-vX', uY-vY', uZ-v7J, 
where u, v have the values given in No. 10, viz. 
^^=-.3Z^(XX'=*+YY'^+ZZ'=“)^-(l+2/^)(Y'Z'X+Z'X'Y+X'Y'Z), 
«;= 3Z^(X^X'+Y^Y'+Z^Z'):-(1+2Z^)(YZX' +ZXY' +XYZ'); 
these values give 
«X-^^X=-3^^{2X^X'^+(XY'+X'Y)YY^-(XZ'+X'Z)ZZ'} 
+(1+2Z^}{(XY'+XYXXZ'+X'Z)+XX'(YZ'+Y'Z} ; 
and therefore 
MX-»X'=-3pf2a’-|/34+(l+2Z»)|p/3y-h“} 
and consequently, omitting the constant factor, the coordinates of F may be taken to be 
17. The line through two consecutive positions of the point F is the line EF. 
The coordinates of the point F are 
— — ?/3"+ya, — ^/+a(3; 
and it has been shown that the quantities a, j3, y satisfy the equation 
-Xa^+/3^+y^) + (-l + 4^>/3y=0. 
Hence, considering a, j3, y as variable parameters connected by this equation, the equa- 
tion of the line through two consecutive positions of the point F is 
_ 3Za^ -f ( - 1 -f 4Z^)/3y, - 3^/3^-f ( - 1 + 4^«)ya, 
-3?/-l-(-l-l-4Z>i3 
— 2lu , y , 
/3 
7 , -2//3 
a 
2, 
/3 , a , 
— 2ly 
and representing this equation by 
we 
find 
Lr-j- fi- N 2 = 0, 
L=(4«’(3y-oi’)(-3;«*+(-l+4?)^y) 
+(«(3+2?/)(-3;^-+(-l+4/V) 
+(«y+2;^')(-3;/+(-l+4P)a|3); 
or multiplying out and collecting, 
L=3lu^+(-l-8Py(3y+(-5l+8l*)(a(B^+uf)+(-16P+16l^)(3y; 
MDCCCLVII. 3 K 
