203] 



GRAPHICAL DETERMINATION OF THE ROOTS. 



339 



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

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

 /3 increases beyond this critical value, one value of x becomes negative, and 

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



Hence, as /3 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. 



(5 = 2 



{3=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 

 /3 we may have one, but never more than one, positive wave for which 

 o- is numerically less than 2n. Finally, when /3 is very great, the value of cr 

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

 remaining values tend all to become more and more nearly equal to 2n. 



If we use a zero suffix to distinguish the case of n=Q, we find 



<V 



(vi), 



222 



