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