326 H. XAGAOKA; DIFFRACTION PHENOMENA 



shows that for large values of x, J^x) must be very small ; thus the 

 abscissae of inflexion points belonging to this set will ultimately be 

 coincident with the roots of 



J u (x) = 0. 

 These inflexion points are nearly coincident with the maximum 

 or minimum position of J x (x). If in the above equation J (x) and J^x) 

 be expanded in semi-convergent series, we get the equation 



^(^-T) = n^- - 5392 ¥ +0 - 603 ¥- 



The ordinates of these inflexion points are given by 



We shall distinguish the inflexion points given by the roots of 



J (x) 

 j^x)z=^o from those given by -77-f = 3 by calling the former inflexion 



points of the first set and the latter those of the second set. The above 

 considerations show that the inflexion points of the second set lie 

 nearly midway between those of the first set, the approximation 

 becoming closer for increasing values of x. 



Between the inflexion point* of the two sets, there must be points 

 of maximum curvature ; these points are evidently given by 



, c Jlx)J x \x) „ J x \x) 

 + J0 x 3 ° x* _U 



If x be large, the leading term is evidently Jo{x) — Ji\x), while the 

 remaining terms decrease very rapidly with increasing values of x ; 

 thus the points of maximum curvature will be approximately given by 



J (x)=±J 1 (x) 



This equation shows that these points lie nearly midway between 

 the inflexion points of the first and of the second set. Thus, if the 

 successive inflexion points of the first set be joined by a series of 



