414] 
ON POLYZOMAL CURVES. 
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equation —'TT* r “r — u gives —r r i r t — djllu eumuiumg wiuii ^t^tu w, 
a u c cl a D c q 
these are only satisfied by one of the systems (a + b = 0, c + d = 0), (a + c = 0, b + d = 0), 
(a + d = 0, b + c = 0). Selecting to fix the ideas the first of these, or writing 
(a, b, c, d) = (a, -a, c, -c), 
so that we have identically 
a (A 0 — B°) -f c (0° — D°) = 0, 
an equation which signifies that the radical axis of the circles A, B is also the 
radical axis of the circles C, D; then, writing as we may do, 
we have 
c’ 2 c ’ 
= 1 + 1, = 2, = 1 — 1, = 0. 
Here fl 1 + fm 1 — fn 1 = 0, which gives one of the trizomals a cubic, viz., this is the 
trizomal 
V cj \ c / 
The other trizomal reduces itself to the bizomal V3l° + V23° = 0, which regarded as a 
trizomal, or written under the form (V2P + V33 0 ) 2 = 0, is the line 31° — 23 =0 twice, viz., 
this is the radical axis of the circles A 1} B x twice; and the order is thus = 2. By 
what precedes, the line in question is in fact the common radical axis of the circles 
A, B and of the circles C, D. 
Article Nos. 203 to 205. Gases of the Decomposable Curve, the Centres in a Line. 
203. We have yet to consider the decomposable case when the centres A, B, G, D 
are on a line; the equation a3l° + b33° + c(S° + dX>° = 0 here subsists universally, what 
ever be the radii a", b'\ c", d”. We establish as before the relation + “ + u = 
U D C Q 
The cases are as follows: 
I. No further relation between l, m, n, p; order of tetrazomal = 8, of trizomals 
4 and 4. 
II. fl + Vm + fn + fp = 0; order of tetrazomal = 7 ; of trizomals = 4 and 3 ; same 
as II. supra. 
III. fl + fp=0, Vm + Vw = 0; order of tetrazomal =6; of trizomals 3 and 3; 
same as III. supra. 
C. VI. 
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