PROFESSOR CAYLEY ON POLYZOMAL CURVES. 



95 



204. IV. Jl + J m + Jn + Jp - 0, a, J I + b J m + cjn + &Jp = 0; order 

 of tetrazomal = 6 ; this is a remarkable case, the orders of the trizomals are 

 either 3, 3 or else 4, 2. 



To explain how this is, it is to be noticed that in the absence of any special 



relation between the radii, the above conditions combined with - + ^ + -+^ = 



abed 



give Jl : Jm : Jn'- Jp = a : b : c : d*; when I, m, n,p have these values, the 

 case is the same as IV. supra, and the orders of the trizomals are 3, 3. But if 

 the radii of the circles satisfy the condition 



1 , 



1 , 



1 , 



1 



a , 



b , 



c , 



d 



a\ 



b\ 



c 2 , 



d 2 



a"\ 



V'\ 



c"\ 



d" 2 



then the two conditions satisfy of themselves the remaining condition 



-+ir+-+^ = 0, and the ratios Jl: Jm : Jn: Jp instead of being deter- 



minate as above, depend on an arbitrary parameter. 

 We have 



Jl x = Jl + fiJY' */»i= Jm — <v bcd7 k ^ n > J n \ ~ J' n + v bed 7 ° ^ m ' 

 and between /, m, n, p only the relations 



J I + Jm + Jn + Jp = , a Jm + b Jm + c Jn + d Jp = . 



We find first 



v/j + Jmj + Jn 1 = Jl + Jm + Jn 



Jp 



+ Jj{l ^p-Jm( h ^ n - c ^'")} 





Jp) - J -j^ (b Jn - c Jm) j , 



* Writing x 2 , y 2 , z 2 , w 2 in place of Ji } J m> J n , Jp, we have to find x, y, z, w from the 

 conditions 



x -\- y + z + w =0, 

 ax 4- by -\- cz -+- dw = , 



abed 



where the constants are connected by the relation 



aa + 6b + cc + dd — . 



It readily appears that the line represented by the first two equations touches the quadric surface in the 

 point x : y : z : w = a : b : c : d , so that these are in general the only values of JJ: J m : J n : Jp . 

 In the case next referred to in the text the line lies in the surface, and the values are not determined. 



