of Edinburgh, Session 1870 - 71 . 
437 
When AP • BC = AO . BQ the curve becomes an ellipse or 
hyperbola. Of this the simplest case is 
AP = BQ> BO = CA. 
The normal at B is in all cases parallel to 
AP . BC • U(BR) =fc AO . BQ . U(AR) , 
because we have 
d . AR = d . BR . 
But the general equation (1), on account of the identity 
AP .BC.BQrbAO.BQ.AP = db AP -BQ . AB , 
may be written more simply, as 
AP.BC.RQ - AC.BQ.PR = 0, (2) 
a very singular and suggestive form ; holding true, as it does, for 
all four points, R, R', R", R'", in the figure. 
Hence the normal is 
U(BR) t U(AR) 
RQ PR ’ 
which may he constructed by drawing at R a tangent to the circle 
circumscribing the triangle PQR. When the curve is a conic this 
line is parallel to CPQ, because by the condition above we have in 
this case 
RQ = PR. 
Of course the mode of tracing here adopted is at once capable of 
being effected mechanically. 
The results above are easily derived from the general equation 
of Cartesian Ovals 
er d= e'r' = a , 
by writing it in the form 
e(r 0 4 - e'x) rh e'(r Q ' ex) — a , 
and showing from this that QP cuts AB in a fixed point. 
But by a purely quaternion process it is easy to give in a very 
simple form the equation of the locus of R when C is not in the line 
AB. Let CA, CB, OR be denoted by a, /3 , p respectively, and let 
3 N 
VOL. VII. 
