ON POLYZOMAL CURVES. 
555 
414] 
there is here a single branch containing (z 2 = 0) the line infinity twice; the order is 
= 6. Each of the points I, J is a double point, and there are therefore two more 
points at infinity, that is ^besides the asymptotes at /, J), there are two (real or 
imaginary) asymptotes. The number of nodes, as in the general case, is = 6. Hence 
dps. are 6+2.1, = 8 ; class is = 14 ; deficiency = 2. 
I notice the included particular case where the circles reduce themselves to their 
centres; viz., we have here the curve 
a V2l + bV23 + cV(S + dV$) = 0, 
which (see ante No. 93) is in fact the curve which is the locus of the foci of the 
conics which pass through the four points A, B, C, D. It is at present assumed that 
the four points are not a circle; this case will be considered post No. 199. If we 
have BG, AD meeting in R\ GA, BD in S, and AB, CD in T, then these points 
R, S, T are three of the six nodes. In fact, writing down the equations of the two 
circles 
b V23 + c V(£ = 0, a V21 + d V3) = 0, 
and observing that when the current point is taken at R, we have 33 : Qi = RB 2 : RC 2 
= (BAD) 2 : (GAD) 2 = c 2 : b 2 , and similarly 21 : 3) = RA 2 : RD 2 = (ABC) 2 : (DBG) 2 = d 2 : a 2 , 
we see that each of the two circles passes through the point R, or this point is a 
node. Similarly, the points S and T are each of them a node. 
V. if 
fl = Vm = V?? = Vp, 
there are here three branches, each ideally containing (2 = 0) the line infinity; the 
order is thus = 5. Each of the points /, J is an ordinary point on the curve; there 
are besides at infinity three points, all real, or one real and two imaginary; that is 
(besides the asymptotes at I, J) there are three asymptotes, all real, or one real and 
two imaginary. Each of the circles V2l + V23 = 0, &c., contains the line infinity, and is 
thus reduced to a line; the number of nodes is therefore = 3. Hence also, dps. = 3; 
class = 14 ; deficiency = 3. 
Article No. 196. Gases of the Indecomposable Curve, the Centres being in a Line. 
196. There are some peculiarities in the case where the centres A, B, C, D are 
on a line; taking as usual (a, b, c, d) for the ¿c-coordinates or distances of the four 
centres from a fixed point on the line, I enumerate the cases as follows: 
I. No relation between l, m, n, p; corresponds to I. supra. 
II. Vi+Vm+Vii + Vp = 0; corresponds to II. supra. 
III. V7+Vm = 0, V?i + Vp = 0; corresponds to III. supra. 
70—2