90 PROFESSOR CAYLEY ON POLYZOMAL CURVES. 



Cases of the Decomposable Curve, Centres not in a line — Art. Nos. 198 to 203. 



198. I assume, in the first instance, that the centres of the circles are not in 

 a line ; we have the following cases : — 



I. No further relation between /, m, n, j> ; the order of the tetrazomal is = 8 

 the order of each of the trizomals is = 4, that is each of them is a bicircular 

 quartic. 



II- J I + Jm + Jn + sfp = ; the order of the tetrazomal is = 7, that of 

 one of the trizomals must be = 3. 



To verify this, observe that we have 



JL + Jm~, + Jn, = Jl + Jm + Jn + -^ + ^E J-^- (c Jm~ b Jn) , 

 1 ' a Jl J I N bed 



or substituting for Jl + Jm + Jn the value — JJ>^ this is 



= <qi { &J¥ ~ dJJ + Vi- {cj7 " 1 " 1>v//7) } ' 



and similarly for JT t + Jm 2 + Jn~ 2 , tne only change being in the sign of the 



fad 

 bc~ 



radical J^-- . But from the two conditions satisfied by I, m, n, p it is easy to 



(*Jp-&Jl? - f- (c Jm ~ hjnf = , 



deduce 



bc 

 and hence one or other of the two functions 



Jl\ + Jm x + Jn t > JJ 2 + Jm 2 + Jn 2 is = ; 



that is, one of the trizomal curves is a cubic. 



III. Jl + Jp = 0, slm+ sin = 0; order of the tetrazomal is = 6; and 

 hence order of each of the trizomals is = 3. To verify this, observe that here 



<(a+-dMH)=° 



which since a + b + c+d = 0, gives — = j— ; so that properly fixing the sign 

 of the radical, we may write Jl + J^- Jm = . We have then 



Jh= nj[- ^. ^ + J* = J^- c ( b + c) Vm ; 



