203] 



GRAPHICAL DETERMINATION OF THE ROOTS. 



339 



As /3 increases, the two values of a- forming a pair become unequal in 

 magnitude, and the corresponding values of x separate, that being the greater 

 for which o-/2?i is positive. When /3=s (a + 1) the curve (iv) and the parabola 

 (v) touch at the point (0, - 1), the corresponding value of o- being -2n. As 

 ft increases beyond this critical value, one value of x becomes negative, and 

 the corresponding (negative) value of cr/2n becomes smaller and smaller. 



Hence, as ft increases from zero, the relative angular velocity becomes 

 greater for a negative than for a positive wave of (approximately) the same 

 type ; moreover the value of o- for a negative wave is always greater than 2n. 



(3-6 



= 40 



9-3 



26-4 



45-0 



70-9 



As the rotation increases, the two kinds of wave become more and more 

 distinct in character as well as in * speed. With a sufficiently great value of 

 ft we may have one, but never more than one, positive wave for which 

 &amp;lt;r is numerically less than 2w. Finally, when ft is very great, the value of &amp;lt;r 

 corresponding to this wave becomes very small compared with n, whilst the 

 remaining values tend all to become more and more nearly equal to 



If we use a zero suffix to distinguish the case of ^=0, we find 



222 



