so that the stability of this axial movement is secured if
| A = 4 | p′ − p | F2 + 4 | q′ − q | FG − G2 |
| r | r |
(25)
is negative, and then the axis makes r√(-A)/π nutations per second. Otherwise, if A is positive
| rt = ∫ | dy |
| y √ (A + 2By + Cy2) |
| = | 1 | sh−1 | √ A √ (A + 2By + Cy2) | = | 1 | ch−1 | A + By | , | ||||
| √ A | ch−1 | y√ (B2 ~ AC) | √A | sh−1 | y √ (B2 ~ AC) |
(26)
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 ½θ = x,
(Nx eψi)3/2 = (x + 1) √ X1 + i (x − 1) √ X2,