| = | r'(s ± p) |
| (r' ± r)² |
Since r' (the speed of the disc) is always positive, and s is always greater than p (cf. [p. 173]), and since the denominator is a square and therefore positive, it follows that
DτW > 0
or that W increases if r increases.
Furthermore, if W is a wide band, s is the wider sector. The rate of increase of W as r increases is
| DτW = | r'(s ± p) |
| (r' ± r)² |
which is larger if s is larger (s and r being always positive). That is, as r increases, 'broad bands widen relatively more than narrow ones.'
3. Thirdly ([p. 174], No. 3), "The width of The bands increases if the speed of the revolving disc decreases." This speed is r'. That the observed fact is equally true of the geometrical bands is clear from inspection, since in
| W = | rs - pr' , |
| r' ± r |
as r' decreases, the denominator of the right-hand member decreases while the numerator increases.