ME. A. CAYLEY ON THE ANALYTICAL THEOEY OF THE CONIC. 
667 
we may write 
(i+p)x=(2-«K -/3y +4(Af'+H^+Gr)(gv+,y+fy). 
(i+p)Y= -„v+(2-/3')y-y'2' +g(H4'+By+F?')(ey+';y+?y). 
( 1 +P)Z= aV -/3''y+(2-y'y+K(Gf'+F^+C?'X|V+>iy+?V), 
where P, which is arbitrary, may be put 
=K(A, ..If', rl, K'f- 
31. These equations then give 
(^,y, ^')=( «, /3, y IX,Y, Z), 
/3', y' j 
which can be verified without difficulty by reversing the process ; and we have thus the 
coordinates (X, Y, Z) in terms of {x\ y\ and reciprocally. 
32. If (Xp Yp ZJ are the coordinates of the other ineunt, we have, it is clear, 
(^,y,^') = ( 2-«, -i3, -y XX,,Y,,ZJ; 
2-/3', -y' 
-/3", 2-/ 
or substituting for (.r', y, 2 ') their values in terms of (X, Y, Z), 
(2X^, 2Y,, 2Zj=( cc, (3, y JX+X^, Y+Y,, Z+Zj, 
a', f3' , y 
i3", y' 
so that (X+Xp Y+Yp Z + Z^) are the same linear functions of 2Xp 2Yp 2Z^ that 
(X, Y, Z) are of (^r', y', z ') ; that is, we have 
i(i+p)(x+x,)=(2-«)x- (3 Y,- y z,+i(Ar+w+Gf')(ry+vy+rA 
i(H-p)(x+x,)= x,+(2-/3')Y- y z,+i(Hr+By+F?')(r^'+,y+yv), 
i(l+P)(X+X,)= -c«" X- /3" Y,+(2-/)Z,+|(G?+F,'+C?')(r3;'+»y+rP). 
which equations may be written 
a+PXX, Y, Z)=:( a, b, c XXpYpZ,), 
a' , b' , c' 
a", b", c" 
4 u 2 
