414] 
ON POLYZOMAL CURVES. 
497 
coefficients l, vn, n are not given in the first instance, but are regarded as arbitrary, 
then the last-mentioned conic is any conic whatever through the three centres, and 
there belongs to such conic and the series of zomals derived therefrom as above, a 
trizomal curve VZ2l° + Vm^B 0 + \/nd° — 0. This is obviously the theorem, that if a 
variable circle has its centre on a given conic, and cuts at right angles a given circle, 
then the envelope of the variable circle is a trizomal curve Vi21° + Vm23° + Vmf°, 
where 21° = 0, 23° = 0, (5° = 0 are any three circles, positions of the variable circle, and 
l, m, n are constant quantities depending on the selected three circles. 
Part II. (Nos. 57 to 104). Subsidiary Investigations. 
Article Nos. 57 and 58. Preliminary Remarks. 
57. We have just been led to consider the conics which pass through two given 
points. There is no real loss of generality in taking these to be the circular points 
at infinity, or say the points I, J—viz., every theorem which in anywise explicitly or 
implicitly relates to these two points, may, without the necessity of any change in the 
statement thereof, be understood as a theorem relating instead to any two points P, Q. 
I call to mind that a circle is a conic passing through the two points I, J, and 
that lines at right angles to each other are lines harmonically related to the pair of 
lines from their intersection to the points /, J respectively, so that when (/, J) are 
replaced by any two given points whatever, the expression a circle must be understood 
to mean a conic passing through the two given points ; and in speaking of lines at 
right angles to each other, it must be understood that we mean lines harmonically 
related to the pair of lines from their intersection to the two given points respectively. 
For instance, the theorem that the Jacobian of any three circles is their orthotomic 
circle, will mean that the Jacobian of any three conics which each of them passes 
through the two given points is the orthotomic conic through the same two points, 
that is, the conic such that at each of its intersections with any one of the three 
conics, the two tangents are harmonically related to the pair of lines from this inter 
section to the two given points respectively. Such extended interpretation of any 
theorem is applicable even to the theorems which involve distances or angles—viz., the 
terms “ distance ” and “ angle ” have a determinate signification when interpreted in 
reference (not to the circular points at infinity, but instead thereof) to any two given 
points whatever (see as to this my “ Sixth Memoir on Quantics,” Nos. 220, et seq.). 
Phil. Trans., voi. cxlix. (1859), pp. 61—90; see p. 86; [158]. And this being so, the theorem 
can, without change in the statement thereof, be understood as referring to the two 
given points. 
58. I say then that any theorem (referring explicitly or implicitly) to the circular 
points at infinity I, J, may be understood as a theorem referring instead to any two 
given points. We might of course give the theorems in the first instance in terms 
explicitly referring to the two given points—(viz., instead of a circle, speak of a conic 
through the two given points, and so in other instances) ; but, as just explained, this 
is not really more general, and the theorems would be given in a less concise and 
c. vi. 63