30 PROFESSOR CAYLEY ON POLYZOMAL CURVES. 



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 6 = 0, <£ = 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 anatytical and partly geometrical, which 

 will be made use of in the following Parts III. and IV. of the Memoir. 



The Circular Points at Infinity ; Itcctanyuhir and Circular Co-ordinates — Art, "Nos. 59 to 62. 



59. The co-ordinates made use of (except in the cases where the general 

 trilinear co-ordinates (x,y,z), or any other co-ordinates, are explicitly referred to), 

 will be either the ordinary rectangular co-ordinates x, y, or else, as we may term 

 them, the circular co-ordinates £, >; { — x + iy, x — iy respectively, i = J — i as 

 usual), but in either case I shall introduce for homogeneity the co-ordinate z, it 

 being understood that this co-ordinate 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 con- 

 cise terms, we may say that the co-ordinates are either the rectangular co-ordi- 

 nates #, y, and z ( = 1), or else the circular co-ordinates £, *i, and «( = !). The 

 equation of the line infinity is z = ; the points I, J are given by the equations 

 (# 4. iy = 0, z = 0) and (x — iy = 0, z = 0), or, what is the same thing, by the 

 equations (£ = 0, z = 0) and (>? = 0, z = 0) respectively; or in the rectangular co- 

 ordinates the co-ordinates of these points are ( — i, 1, 0) and (i, 1, 0) respectively, 

 and in the circular co-ordinates they are (1, 0, 0) and (0, 1, 0) respectively. It is 

 of course, only for points at infinity that the co-ordinate z is = (and observe that 

 for any such point the x and y or £ and ij co-ordinates may be regarded as finite) . 

 for every point whatever not at infinity the co-ordinate z is, as stated above, = 1. 



60. Consider a point A, whose co-ordinates (rectangular) are (a, a', 1) and 

 (circular) (a, a', 1), viz., = 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(g — a'z) = 

 respectively, or they are 



I — az = , >j — a'z = 



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 /, «/, respectively. 



61. We have from either of the equations in (x,y,z) 



(x — az) 2 + (y — a'z) 2 = 0, 



that is, the distance from each other of any two points (x,y, 1), and (a, a, 1) in a 



