If BQ be the direction for the first minimum (the darkness between the central and first lateral band), the relative retardation of the extreme rays is (mn+1)X.
Suppose now that X+SX is the wave-length for which BQ gives the principal maximum, then (mn+1)A=mn(X+SX); whence OX/X= limn.
The equation of the latter, referred to its principal axes, being as in II (41), the co-ordinates of the point J where it is met by the instantaneous axis are proportional to p, q, r, and the direction-cosines of the normal at J are therefore proportional to Ap, Bq, Cr, or X, u, v.
If we now apply them to the case of a rigid body moving about a fixed point 0, and make Ox, Oy, Oz coincide with the principal axes of inertia at 0, we have X, u, v=Ap, Bq, Cr, whence A (B C) qr = L,
To show the cause of this motion, let BQ represent a section of an oblate spheroid through its shortest axis, PP. We may consider this spheroid to be that of the earth, the ellipticity being greatly exaggerated.