562 
ON POLYZOMAL CURVES. 
[414 
204. IV. Vi + Vm + Vn, + Vp = 0, a Vi + b Vm + c V?i +d Vp = 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 -+^+-+^=0 give 
£1 D C Cl 
Vi : Vm : Vw : Vp = a : b : c : d( J ); 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? , 
c 2 , 
d 2 
a" 2 , 
b"\ 
c" 2 , 
d" 2 
then the two conditions satisfy of themselves the remaining condition 
l m n p 
- + v + -+5 = °> 
abed 
and the ratios Vi : Vm : Vn : Vp instead of being determinate as above, depend on an 
arbitrary parameter. 
We have 
Vm! = Vm — 
— - b Vw 
bed l ’ 
V 
n, = \!n + \J 
bed 
and between i, m, n, p only the relations 
Vi + Vm + Vii + Vp = 0, a Vi + 6 Vm + c Vw -f ci Vp = 0. 
We find first 
Vij + Vm 1 + Vw 1 = Vi + Vm + Vw 
+ v?{ a vÿ ~Vra (W ’ i - oVm) } 
= -^ |g(dVi-a V^)-V^;( b ' / »-c' / ™)j. 
1 Writing X-, y' 2 , z 2 , id 2 in place of Vi, Vm, Jn, Vp, we have to find x, y, z, w from the conditions 
x + y + z + w = 0, 
ax + by +cz + dw = 0, 
where the constants are connected by the relation 
aa + 6b + cc + <Zd = 0. 
It readily appears that the line represented by the first two equations touches the quadric surface in the point 
x : y : z : ic = & : b : c : d, so that these are in general the only values of Vi : V/a : Va : Vp". In the case next 
referred to in the text the line lies in the surface, and the values are not determined.