544 
ON POLYZOMAL CURVES. 
[414 
four sets of concyclic foci, there are four different constructions for the curve, viz., the 
orthotomic circle may be any one of the four circles 0, R, S, T, which contain the four 
sets of concyclic foci respectively ; and the conic of centres is a conic through the 
corresponding set of four concyclic foci. We have thus four conics, but the foci of each 
of them coincide with the nodo-foci of the curve, that is, the conics are confocal ; that 
such confocal conics exist has been shown, ante, Nos. 78 to 80. 
Article Nos. 180 and 181. Remark as to the Construction of the Symmetrical Curve. 
180. It is to be observed that in applying as above the theorem of the variable 
zomal to the construction of a symmetrical curve, the orthotomic circle made use of 
was one of the circles R, S, T, not the circle 0, which is in this case the axis; in 
fact, we should then have the conic and the orthotomic circle each of them coinciding 
with the axis. And the variable circle, qua circle having its centre on the axis, cuts 
the axis at right angles whatever the radius may be; that is, the variable circle is 
no longer sufficiently determined by the theorem. The curve may nevertheless be 
constructed as the envelope of a variable circle having its centre on the axis; viz., 
writing 21° —(x — az) 2 4- y 2 — a" 2 z 2 , &c., and starting with the form 
fW + 'JmW + \4r = 0, 
then recurring to the demonstration of the theorem (ante, No. 47), the equation of 
the variable circle is a2l 4- b53 +cQ>°=0, where a, b, c are any quantities satisfying 
- + 4- ~ = 0, or, what is the same thing, taking q an arbitrary parameter, and writing 
^ = 1 4- q, ^ = 1 — q, ^ = — 2, the equation of the variable circle is 
_J_ /21° + _L m <B 0 _ £ W (£° = o. 
1+9 1-q 2 
Compare Nos. 118—123 for the like mode of construction of a conic; but it is proper 
to consider this in a somewhat different form. 
181. Assume that the equation of the variable circle is 
£>° = (x — dz) 2 4- y- — d" 2 z 2 = 0 ; 
we have therefore identically 
viz., this gives 
a2P 4- b$3° 4- c(£° 4- d£)° = 0, 
a 4-b 4-c = — d , 
aa 4- bb 4- cc= — dd, 
a (a 2 - a" 2 ) 4- b (6 2 - h" 2 ) + c (c 2 - c" 2 ) = - d (d 2 - d" 2 ), 
and from these equations we obtain a, b, c equal respectively to given multiples of d ; 
substituting these values in the equation -4-t+ - = 0, d divides out, and we have an 
a b c