498 
ON POLYZOMAL CURVES. 
[414 
familiar form. It would not, on the face of the investigations, be apparent that in 
treating of the polyzomal curves 
VZ (© + 7<P) + Vm (© + M$>) + &c. = 0, 
(0 = 0 a conic, d? = 0 a line, as above), that we were really treating of the curves the 
zomals whereof are circles, and therein of the theories of foci and focofoci as about 
to be explained. And for these reasons I shall consider the two points © = 0, <4> = 0, 
to be the circular points at infinity 7, J, and in the investigations, &c., make use of 
the terms circle, right angles, &c., which, in their ordinary significations, have implicit 
reference to these two points. 
The present Part does not explicitly relate to the theory of polyzomal curves, but 
contains a series of researches, partly analytical and partly geometrical, which will be 
made use of in the following Parts III. and IV. of the Memoir. 
Article Nos. 59 to 62. The Circular Points at Infinity ; Rectangular and Circular 
Coordinates. 
59. The coordinates made use of (except in the cases where the general trilinear 
coordinates (x, y, z), or any other coordinates, are explicitly referred to), will be either 
the ordinary rectangular coordinates x, y, or else, as we may term them, the circular 
coordinates £, g (= x + iy, x — iy respectively, i = V — 1 as usual), but in either case I 
shall introduce for homogeneity the coordinate z, it being understood that this 
coordinate is in fact = 1, and that it may be retained or replaced by this its value, 
in different investigations or stages of the same investigation, as may for the time 
being be most convenient. In more concise terms, we may say that the coordinates 
are either the rectangular coordinates x, y, and z (=1), or else the circular coordinates 
£, 7), and z {— 1). The equation of the line infinity is z = 0] the points 7, J are given 
by the equations {x + iy = 0, z — 0) and {x — iy = 0, 2 = 0), or, what is the same thing, 
by the equations (f = 0, z = 0) and (77 = 0, z = 0) respectively ; or in the rectangular 
coordinates the coordinates of these points are (— i, 1, 0) and (i, 1, 0) respectively, and 
in the circular coordinates they are (1, 0, 0) and (0, 1, 0) respectively. It is, of course, 
only for points at infinity that the coordinate 0 is = 0 (and observe that for any such 
point the x and y or £ and 77 coordinates may be regarded as finite) ; for every point 
whatever not at infinity the coordinate z is, as stated above, = 1. 
60. Consider a point A, whose coordinates (rectangular) are (a, a', 1) and (circular) 
(a, a', 1), viz., a = a + a'i, a' = a — a'i ; then the equations of the lines through A to 
the points 7, J, are 
x — az + i(y — a'z) =0, x— az — i (y— a'z) = 0 
respectively, or they are 
^ — az = 0 , rj — a'z = 0 
respectively. These equations, if (a, a') or (a, a!) are arbitrary, will, it is clear, be the 
equations of any two lines through the points 7, J, respectively.