286 



Abscissa n.6 + a O 



G. B. AIRY ESQ. ON A PROOF OF THE THEOREM 



P. curves only. 



P(0) 



2r 



I 



Abscissa n6 + a o 



Q- curves only. 



, QiO) 



■---.?( <^) 



p. curves and Q- curves combined. 



2ir 



7. In these diagrams it is to be remarked, 



(1) That P (0) and Q (O) do not intersect. 



(2) That P(co) and Q ( 05 ) intersect in two points. 



(3) That one of these points of intersection is above the line of abscissa, and 



the other is below it. 



Let us now consider what must have happened in the change of relation of the P-curve 

 and the Q-curve, while P (0) was changing to P ( co ) and <2 (o) to Q ( co ). 



8. We have preserved no record of the forms of the curves for values of r inter- 

 mediate between and 00 , but we know that they will depend upon the terms of the 

 expressions for P and Q which are intermediate between their first and last terms, and 

 therefore will be different in different cases. But we see that some of the following 

 Changes must have occurred in every case, while r was increasing from to co ; — 



Change [ll. Either a sinus of the P-curve must have intruded upon the Q-curve 

 (or vice versa) so as to produce two intersections. 



