Fy3 - Gx3 + i[root]X3
F(X + Yi) = --------------------- [epsilon]^[psi]_i. (22)
[root](F² - x3²)
Suppose x3 - F is a repeated factor of X3, then y3 = G, and
_ _
| p´ - p q´ - q |
X3 = (x3 - F)² | ------(x3 + F)² + 2------G(x3 + F) - G² |, (23)
|_ r r _|
and putting x3 - F = y,
_
/ dy \² | p´ - p q´ - q
( -- ) = r²y² | 4 ------ F² + 4 ------ FG - G²
\ dt / |_ r r
_
/ p´ - p q´ - q \ p´ - p |
+ 2 ( 2 ------ F + ------ G ) y + ------ y² |, (24)
\ r r / r _|
so that the stability of this axial movement is secured if
p´ - p q´ - q
A = 4 ------F² + 4 ------FG - G² (25)
r r
is negative, and then the axis makes r[V-](-A)/[pi] nutations per
second. Otherwise, if A is positive
_
/ dy
rt = | ----------------------
_/ y[root](A + 2By + Cy²)
1 sh^(-1) [root]A[root](A + 2By + Cy²) 1 ch^(-1) A + By
= ------- ---------------------------- = ------- ---------------, (26)
[root]A ch^(-1) y[root](B² ~ AC) [root]A sh^(-1) y[root](B² ~ AC)
and the axis falls away ultimately from its original direction.
A number of cases are worked out in the American Journal of
Mathematics (1907), in which the motion is made algebraical by the use
of the pseudo-elliptic integral. To give a simple instance, changing
to the stereographic projection by putting tan ½[theta] = x,
(Nxe[psi]i)^3/2 = (x + 1)[root]X1 + i(x - 1)[root]X2, (27)
X1
-- = ±ax^4 + 2ax³ ± 3(a + b)x² + 2bx ± b, (28)
X2