68 
ME. A. CAYLEY’S SIXTH MEjMOIE TPOX QrAXTICS. 
the equation of any 
ways, in the form 
other point-pair whatever can be expressed, 
(«, cX^, =0 ; 
and that in two different 
the points (te+my=0) corresponding to the two admissible values of the linear func- 
tion being in fact the harmonics of the point-pair in respect to the given point-pan 
(a b cTx vY=0, or what is the same thing, the sibiconjugate pomts of the miolunon 
formed bi the two point-pairs (see Fifth Memoh, No. 10.5). The point-pair represented 
by the equation in question does not in itself stand in any pecuhar relaUon to the given 
point pail- (a, b, cjx, y)*=0 ; but when thus represented it is said to be mscnbed m t e 
given point-pair, and the point lx+m^=0 is said to be the axis of inscmption -tod the 
harmonic of this point with respect to the given point-pau- (that is, the other sibiconjugate 
point of the involution of the two point-pairs) is said to be the centie of mscnption . 
166. We may, if we please, and 6 being constants, exhibit the equation of the 
inscribed point-pair in the form 
(a, b, cite, yf{a, b, ijf sin» «-(<«;- 5»)(.ry-.Vy)*=0, 
where we have for the axis of inscription and centre of inscription respectively, the 
equations „ 
xy —xy ' -“'J? 
(«, h,cjx, yjx\ y)=0. 
Or in the equivalent form, 
{a, b, cjx, y)\a, b, ejx!, cos* b, cjx, yli', y)}’=0, 
where we have for the axis of inscription and the centre of inscription respectively, the 
equations y)=0, 
xy' — x'y =0. 
167. The equivalence of the two forms depends on the identical equation 
(«, b, cXx, yf{a, i, yf- {(«> A*’ A-f'- f)r=(ac-V-txy‘-Yy)\ 
which is in fact the equation mentioned. Fifth Memoir, No. 95. If, for shortness, we write 
(a,J,cXai,y)*=00, 
(a, 5, cX^, yX!i\ y) = 01 = 10, 
&c.. 
then the equation may be represented in the form 
00, 01 
z=[ac—lf) 
x,y 
10, 11 
X, y' 
168. There is a like equation for the three sets {x,y), (x', y'), {x", y")\ the right- lan 
side here vanishes, for there are not columns enough to form therewith a deterniinaut. 
* The words inscribed, inscription, are used not in opposition to, but as identical with, the woid. d 
cumscribed, circumscription; and in like manner {fost, Nos. 203 et seq.) as regards conics. 
