910 REPORT—1885. 
The orthoptic locus of the cardioid, a curve of the third class, is known to consist 
of a circle and a bicircular quartic, together making up a ¢rictreular sextic. Such 
likewise must be the locus for a curve of the third class in general, since three 
tangents can be drawn to it from I (or J), and they may be regarded as intersect- 
‘ing two and two at right angles. Consequently I and J are threefold points on the 
locus: and they are its only points at infinity. 
If the original curve be of class ”, then from I (or J) x tangents can be drawn to 
(n= 
+) ways in orthogonal pairs. This being there- 
it, and these can be taken in x — 
fore the order of each of the circular points on the orthoptic locus, the order of the 
locus is 2 (n-1). 
The order of the locus is apparently reduced when the original curve towches the 
line at infinity. Thus, in the parabola the line at infinity may be regarded as at 
right angles to the second tangent that can be drawn from any point upon it. 
Every such point therefore belongs to the locus, and the remainder of it when the 
line IJ is subtracted is a straight line not passing through the circular points, 
evidently the polar of the focus 8, since SI (or SJ) is a tangent at right angles to 
itself. In like manner the reduction for a curve of class x which touches the line 
IJ in » points may be estimated. 
By the same kind of argument it may be shown that if pairs of tangents be 
drawn to two curves of class m and class respectively, each to each, their locus 
of intersection will be a curve of the order 2mn, passing mn times through the 
circular points. The pedal of a curve of class is therefore an n-circular 2n-ic. 
That of an ellipse in general a bicircular quartic. 
It remains to explain the apparent reduction of the order of the pedal with 
respect to a focus. 
Taking for example the case of the ellipse, and eliminating between the 
equations :— 
y — mx = /(b? + ma’) 
and 
my + «= /(a?— 6?) 
we have 
(1 + m*) (a? + y? — a*) =0 
The factor 1 +m? equated to zero gives the directions of I and J, the tangents 
from which intersect in opposite pairs at the foci. The perpendicular from a focus 
S to the tangent SI is SI itself, which is accordingly a factor of the pedal, as like- 
wiseis SJ. But SI and SJ make up the potnt-circle at S. This factor corre- 
sponding to 1+m* being rejected, the remainder of the pedal must be a circle, 
evidently that on the axis as diameter. 
2. On the Reduction of Algebraical Determinants. 
By W. H. L. Russett, F.R.S. 
_ The following method, it will readily be seen, is applicable to all determinants 
in which the constituents of every column are rational and entire functions of (). 
Consider the determinant :-— 
a, + b,a + 0,27 + d,x", a, + b,x + ¢,27 + da, a, + byt + 0,0? + d,2°, 
at bye + Cu? + d,a?, a,+b.v + ¢,27 + da, a, + bv + cv? + d,r', 
. 2 3 , 3 
Ay + b,% + CyX* + dyX®, dg + bgt + Cgt* + dt, ay + byt + C527 + dyr*, 
Divide the first, second, and third rows by a, a, a, respectively, and subtract 
the first row from the second and third rows, and the determinant becomes 
f 1 / ean 12 
1+ b’ e+ e wt dit a’, + bau + cq" + C528, a4 +0/,0 +007 + d’,0°, 
0+ ba 4 ¢/,27 + d/,05, a+ 0/0 + ¢',07 + d’52, a’ +O oa t+ C/gt? + dU u* 
aa oder guiness tee ARONA een a its Pee oq 
040,27 40,07 + d/,2°, a’, + b’4x + c’42” + d’gt®, a’y + b’gu + C2” + d' 42”, 
that is :— 
a’, +02 +¢/,07 + d’,a%, a’, + Ou + cx? + d’,x%, 
a’, + bt + 0/0? +. d’gu°, a’, +O’ gu + c'2? + d’yt°, 
