570 
ON POLYZOMAL CURVES. [-114: 
which is the relation between the distances of any four points on a cone-sphere, this 
•equation may be written under the irrational form 
23.14 + 31.24 + 12.34 = 0. 
Taking (a, a', ct"), (b, b\ b"), (c, c, c"), (x, y, z) for the coordinates of the four points 
respectively, we have 
23 = V(6 - cf + (b' - c') 2 + (b" - c")\ 14 = - a) 2 + (y- aj + (z - a)\ 
31 = V(c - af + (o' - aj + (c" - a") 2 , 24 = \f(x - by + (y- bj + (z- b'J, 
12 = V(a — b) 2 + (a' - bj + (a" - b'J, 34 = \/(r - c^+TF-^F+T^-^y 2 . 
or the symbols having these significations, we have 
23 . 14 + 31. 24+12.34 =0 
for the equation of the cone-sphere through the three points; or rather (since the 
rational equation is of the order 4 in the coordinates {x, y, z)) this is the equation 
nf the pair of cone-spheres through the three given points; and similarly it is in 
the first problem the equation of a pair of circles each touching the three given circles 
respectively. 
In the first problem the radii of the given circles were i (z — a"), i {z — b”), i(z — c") 
respectively; denoting these radii by a, /3, 7, or taking the equations of the given 
circles to be 
(x — of + (y — a') 2 — a 2 =0, 
(x — by + (y — b') 2 — /3 2 = 0, 
(x - c ) 2 + (y - c ) 2 - 7 2 = 0, 
the symbols then are 
23 = V(6 — c) 2 + (b' — c') 2 —(/3 — y) 2 , 14 = V(x — a) 2 + (y — ci') 2 — a 2 , 
31 = V(c — a) 2 + (c — ci) 2 — (7 — ci) 2 , 24 = V(x — b) 2 + (y — b') 2 — /3 2 , 
12 = V(a — bj 2 + (a' — b') 2 — (a — /3) 2 , 34 = \f(x - c) 2 + (y - c') 2 — y 2 , 
and the equation of the pair of circles is as before 
23.14 + 31.24 + 12.34 = 0; 
where it is to be noticed that 23, 31, 12 are the tangential distances of the circles 
2 and 3, 3 and 1, 1 and 2 respectively; viz., if a, /3, 7 are the radii taken positively, 
then these are the direct tangential distances. By taking the radii positively or 
negatively at pleasure, we obtain in all four equations—the tangential distances being 
all direct as above, or else any one is direct, and the other two are inverse; we have 
thus the four pairs of tangent circles. 
The cone-spheres which pass through a given circle are the two spheres which 
have their centres in the two antipoints of the given circle; and it is easy to see 
that the foregoing investigation gives the following (imaginary) construction of the