658 
ME. A. CATLEY ON THE ANALYTICAL THEOEY OE THE CONIC. 
where the values of the coefficients are 
-- ;|g(^x'-A')+|r(A5' +H,' -f 
+B.' + F?') , -/»')+! C?'^-2Gri'-A|'2), _^(5r?:'_cr)+|r(Hr +BV + 
EEx' + CD , -^(*l’-/^x')+|.'(G|’ + FV + CD . l-;^(/|'-yV)+g (C?'S- A|--: 
and considering (X, Y, Z) and (X^ Z,) as quantities connected by the foregoing 
linear relations, we have identically 
{a, ..IX, Y, Zf=(a, ..JX,, Y„ Z,y. 
So that the investigation leads to the automorphic transformation of the quadric func- 
tion, a transformation first effected by M. Heemite*. 
33. It is to be remarked that the foregoing formulae show that (x', y, z') being the 
coordinates of a point on the conic (a, ..yjc, y, zf-\-{^x-\-'^'y-\-^'zy=^^ from which point 
tangents are drawn to the conic («, ..I^, y, 2 )^= 0 , then the coordinates (a/, y\ z') enter 
linearly into the equations of the tangents, the ineunts (or points of contact), and the 
polar. And it may be added that the equation of the conic enveloped by the polar 
(that is, the polar conic of {a, ..I^r, y, zy-\-{^x-\-rly-V^'zy~0) has for its equation 
{K-f(A, ..ir, rf, r)n(«, ..fe y. ;^)^-K(r^+p;'2/+W=0. 
and that the coordinates of the point of contact of the polar with this conic are 
^+e(a? +H,'+G rx?*'+>/y+?'2'). 
y+g(Hr+Bx +Fr)(?v+yy +?*'), 
z'+ifor+Fy +cr)(?*'+yy+fV); 
so that (.r', y, 2 :') also enter linearly into the expressions for the coordinates of the last- 
mentioned point. 
Article Nos. 34 to 37, relating to two conics. 
34. Considering now the two conics 
U=(«, h, c,f hXx, y, 2)"=0, 
U = (a', 5', (/,/', y, 7i'I^, y, zf=^ ; 
Suppose that the conic 
SU-f fl'U'=(Sa-bSV, ..Jx, y, z)^=0 
represents a pair of lines. 
* See my “ Memoir on the Automorphic Transformation of a Bipartite Quadric Function,” Phil. Trans, 
vol. cxlviii. (1858) pp. 39-46. 
