654 
ME. A. CAYLEY ON THE ANALYTICAL THEOEY OE THE CONIC. 
V= X , y , z 
Ar+W+G^, Hr+B;?'+JT, Gr+F;j' + C^' 
_^^').y(Gr +F;j' 4-C?')-2(Hr+BV +Fr ) 
(xr-«'r ) . ^(Ar +H;j'+GO-^(G| ’ + Fpj' + C^' ) 
(A^r-X^' ). 4Hr+B;j' +F?')-3/(A|' +H;j'+Gr) 
=x{(i'^+»j'2/+r2)(Ar +w+Gr)-4A, ..XI', ri, rn 
+^{(|'^+Vy+r2)(H|'+B;j'+Fr)-y(A, ..jr, r!, i;j] 
{(|'^+;?'y+r;s)(G|'+F;j'+Cr)-^(A,..II', n', KJ) 
= (|'^+;j'3/+^';5).(A, .-X^? {^1 *'X^^ ^') — (X^'l-f^3/+<'^)-(A, --X^^’ ^'T 
= K(/|'+?w;?' + 0.(|'^+;j'?/ + ^'2)-(«, --X^^ --X^'’ 
and the formula II. thus becomes 
{a, ..X^\ y, 2 )^=Quotient by (A, ..Xl', K'f of I 
K(|'a.’ 4" ^'y + 
II (bis) +Quotient by (a, ..^l, w, --X^^ — K(/|'+m^j'+’^C)^ of Prodn 
I \¥.{l^+mrl^nt^).{^x-\-rly-{-^z)—{a, m, n\x,y, 2) .(A, . .XI', »?', ^')'l 
L I ±x/-(A, ..XI', ??', ^'KA, . rXjnz-ny , . Xi', ’j', ^')- -I 
26. And from these we at once deduce the inverse or reciprocal formulae 
(A, ..XI, >7, ^)"=Quotient by (A, ..XI', n\ K'f of] 
r 
III. (bis) 
[(A,..xi,^,?Xr,^',nT 
+Quotient by (a, .-X^, (A, • -X^^ ^0^ — K(/|'+W+w?')'^ of K into . 
|(Z|'+m^'+^^D(A, .-X^, ^X^', ?"')— (^I+wi>7+^^)/(A, --X , ?) I 
A±n/-(A, .-X^', . -XZ, w, nXnK'-^'K^ • •), J 
27. And 
(A, . .XI, »7, ^)"=Quotient by («, . -X^', y, of j 
^{x'l^yWKf ' 
^ +Qiiotient by (A, .-X^, *')^*(^, --X^’ X’ 2 ')^— K(X.a?'+y/y'+v 2 ')^ of Produ 
[K(Xa^'-}-j«<y'+t'2').(a7'H-y'^+^'^) — (A, • -X^, ?X^, . •X^’', ^) ]. , 
J ±x/— K(a, . .x^', y, • -X^, y^ z'X(M^~r/i , . .), i I 
which four formulae have the sa.me geometrical significations with the original four j 
formulae to which they correspond respectively. 
