(8)
| (1 − f2) K′ = ∫ −1z1 | √ (z3 − z1) dz |
| √ ( −4Z) |
| = sn−1 √ | −1 − z1 | = cn−1 √ | 1 + z2 | = dn−1 √ | 1 + z3 | . |
| z2 − z1 | z2 − z1 | z3 − z1 |
(9)
Then if v′ = K + (1 − f′)K′i is the parameter corresponding to z = D, we find
f = f2 − f1, f′ = f2 + f1,
(10)
v = v1 + v2, v′ = v1 − v2.
(11)
The most symmetrical treatment of the motion of any point fixed in the top will be found in Klein and Sommerfeld, Theorie des Kreisels, to which the reader is referred for details; four new functions, α, β, γ, δ, are introduced, defined in terms of Euler’s angles, θ, ψ, φ, by