of Edinburgh, Session 1870 - 71 . 
437 
When AP • BC = AC . BQ the curve becomes an ellipse or 
hyperbola. Of this the simplest case is 
AP - BQ, BC = CA - 
The normal at R is in all cases parallel to 
AP. BC • U(BR) =fc AC . BQ . U(AR), 
because we have 
cl . AR = d . BR . 
But the general equation (1), on account of the identity 
AP -BO.BQdbAC.BQ.AP = =bAP.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 /7/ , in the figure. 
Hence the normal is 
TJ(BR) 5 TJ(AR) 
RQ PR ’ 
which may be 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 =±r eV = a . 
by writing it in the form 
e(r 0 + e'x) dh. e'(r 0 '-=f = 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, CR be denoted by a, /3, p respectively, and let 
3 N 
VOL. VII. 
