ATT?. A. CAYLEY ON THE ANALYTICAL THEOEY OE THE CONIC. 
651 
contact: {I, m, n) are supernumerary arbitrary quantities, the values whereof do not 
affect the result And in the formula II. we have 
®}=n+y-(A, ..If, ..I*, y, z) 
(for the value of II see the formula). The quantities ti', are the line-coordinates 
of the chord of contact (viz. the point-equation of this line is ; (X, (a, y) 
are supernumerary arbitrary quantities. 
23. In the like manner the formulae III. and IV. each of them show that the line- 
equation A 
(A, ..II, 0 =0 
of the conic may be written in the form 
W>+iQR=0, 
where Q=0, 11=0 are any two ineunts of the conic, and W=0 is the point of inter- 
section of the corresponding tangents ; viz. in the formula III. we have 
w=(A, ...ir, ». a 
®j = (a, ... I., .)+V-(A,..ir,,',?? 
X, y 
(for another form of Q, E see the formula). 
The quantities yj, are the line-coordinates of the line through the two ineunts, 
or chord of contact ; (A, /a, y) are supernumerary arbitrary quantities ; and so in the 
formula IV. we have 
W=^H-yj?+;sI, 
^|=KO+v^-K(a, ..Jed, y, (A, ..JnJ—mz', ..J^, ri, 
(for the value of O see the formula), where ed, y', ^ are the point-coordinates of the 
intersection of tangents at the two ineunts, or pole of the chord of contact ; {I, m, n) are 
supernumerary arbitrary quantities. 
24. We may, instead of the supernumerary arbitrary quantities {I, m, n) of the 
formula I., introduce the quantities (X, yj, v), where 
{I, m, n)-= 
1 ( A, 
H, 
G, 
H, 
B, 
F, 
G fA, 0- 
F 
* In a different point of view, viz. if we consider the formula I. as a transformation of the function 
(a, ...Jx, y, zy, then (x', y', z') and {I, m, n) would be each of them supernumerary arbitrary quantities : 
and so in the other like cases. 
