278 PEOFESSOE C. J. JOEY ON QUATEENIONS AND PEOJECTIYE GEOMETEY. 
SZ 5 ' = 0 , but subjects such as that just mentioned may be readily investigated by 
writing 
O 
q ■= xa yh zc, q' — x'a -f- y'h 
where a^h and c are any three points in the plane. Then the array 
{qq'\ =\[hc]-\-ix {ca\-\-v {ah] if \=yz'—y'z, y=zx'—zx, v=xy'—x y . ( 327 ) 
and 
^ + y [JiC, fta\ + V [fta, fth\ . . . (328), 
ft = fi -h tf 
Hence (compare (301) and (296)) the line qq' joins a jioint to its correspondent in 
the transformation produced by ft 'i? 
^^^(hcfthftc)+ tyv {(caftafih) + (ahftcfta)} = {) . . . . (329). 
This etjuation may be regarded as the tangential et|uation of a conic involvmg a 
parameter t quadratically. For six values of t the equation represents a pair of 
points—one point of each pair being one of the six points in which the plane meets 
the ciiticai sextic, and the second point being the intersection of the plane with the 
line into which the plane is transformed by the function (ft — s) which destroys the 
aforesaid point (compare i\.rt. 14, I). 
In a united plane, the theory is simpler. Let «, b, c be the united points in the 
plane, united points of yj. Then (327) and (328) become 
[yq ] = X [bo] + y [ca] + v {ah}, 
(/i + t/s)'/] tzC-\-tfy:}-\-ix\tye-\-tfoC, ty:t-\-tfy:i} 
. + + tfy:i, t.Jj + tf.h] . (330); 
and we get the conics 
{bcfj)fyi)^tyv [{caf^afgj)^(abf,cfyi)'\\~a (h-t^) yv {abcfyt) = 0 . (331). 
In this case the system of conics is inscribed to a common quadrilateral. 
The conic enveloped by the satellites is 
or 
SX% (bc^obfoc) + tyv [G (abfyefyi) + (cq/^a/oZi)] = 0.( 332 ). 
80. More particularly for the functions in a united plane of /’ the united 
points a, b, c form a triangle (I) in perspective with the triangle (II) of the traces 
of tlie united planes of the conjugate ; for these planes are 
S( 2 « = 0 , 8176 = 0 , 87 ^ = 0 .( 333 ); 
and the centre of perspective is given by 
S^aSlc = S(^6Sc« = SqcSitb 
( 334 ); 
