488 
ON POLYZOMAL CURVES. 
[414 
the equation becomes 
^(¿d+pa) Vi7+d Vim VF+d ViwVlV^ = ^p(b\/n*JV— c'Jms/w'^j , 
or as this is more conveniently written 
(V^+ -^7|) VF + VmF + VwT) 2 = f (b VVF-cVmr) 2 , 
an equation breaking up into two equations, which may be represented by 
+ Vm a V + V?q W — 0, \/TM + Vm 2 F + Vw 2 W= 0, 
where 
A =Vz +5 4 
d Vi 
Vm, = Vm- /v /^ a fbV; 
Vwx = Vw + /y/j c Vra 
Vi 2 = Vi +5 £= 
d Vi 
v ™ s = v ™ + VbTdf b ' / ” 
Vn 2 = Vw | c ^ 
where, in the expressions for Vi, &c., the signs of the radicals 
Vi Vra Vw a / —- - > 
3 3 3 V bed l 
may be taken determinately in any way whatever at pleasure; the only effect of an 
alteration of sign would in some cases be to interchange the values of (V ii,Vm x , Vwj) 
with those of (Vi 2 , Vm 2 , Vw 2 ). The tetrazomal curve thus breaks up into two trizomals. 
44. It is to be noticed that we have 
L m, n x 
1 + -7- + - = 
a b c 
i , ?£ 2 . 
a dH + d 
m a np 
+ b cd i 
n a mp 
H h rj —j- 
c bd i 
that is 
and that similarly we have 
ii . ph , Py r> . 
T" 1 I U 5 
a b c 
J/n Wio Tin A 
- + -V + — = o. 
a b c