94 PROFESSOR CAYLEY ON POLYZOMAL CURVES. 



(a + b = 0, c + d = 0), (a + c = 0, b + d = 0), (a + d = 0. b + c = 0). Select- 

 ing to fix the ideas the first of these, or writing 



(a,b,c, d) =(a, - a, c, - c), 



so that we have identically 



a(/-£°) + c(C'°- B°)=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, J ^ ? ( — J -a)— 

 we have 



Jm . — 1 + . J in . = 1 — - i 



1 c i c 



J- = 1 + 1 , = 2 , J^ = 1 - 1 , = . 



Here Jj + dm, — dn x = 0» which gives one of the trizomals a cubic, viz.. 

 this is the trizomal 



(l _ *) v/A° + (l + J) v/B 5 + 2 Jc°= 



The other trizomal reduces itself to the bizomal «/A c + ^/gp = 0, which regarded 

 as a trizomal, or written under the form ( d^ + dW) 2 = 0, is the line A°— B°= 

 twice, viz., this is the radical axis of the circles A v 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. 



Cases of the Decomposable Curve, the Cotter* hi a Line — Art. Nos. 203 to 206. 



203. We have yet to consider the decomposable case when the centres 

 A,B, C, D are on a line; the equation aA°+ bB°+ cC°+ dD°= here subsists 

 universally, whatever be the radii a", b", c", d" . We establish as before the 



relation - + ~ + - + ^ = . The cases are as follows : — 



abed 



I. No further relation between /, m, n, p , order of tetrazomal = 8, of trizomals 

 4 and 4. 



II. dl + dm + dn + dp = ; order of tetrazomal = 7 ; of trizomals = 4 and 

 3 ; same as II. supra. 



III. Jl + dp = 0, dm + dn — ; order of tetrazomal = 6 ; of trizomals 

 3 and 3 ; same as III. supra. 



