152 



MR. S. S. HOUGH ON THE APPLICATION OF HARMONIC 



X has a value in the neighbourhood of a root of the equation Lj, = 0, the continued 

 fractions H*_ 2 , K' l+2 will rapidly converge to small values. Further, with large 

 values of n, the numerical values of H_ 2 , Kj l+2 tend to become equal with opposite 

 signs. Hence there will be roots of the equation (46) which approximate to roots of 



the equation 



L* = (47). 



Let us therefore examine the nature of the roots of this latter equation ; this 

 may best be done by considering the graph of the function A*. Putting y = A', 

 x = X/w, we have to consider the form of the curve 



j o " (n + 1) '2s/,c (ii 1)- (n s) (w + s) 



n~(n+ I) 3 



n" ('2n - 1) (2>t + 1) {n (n - I) - 2s/x] 

 ( + 2) 2 (n -s -f l)(u +s+ 



(n + I) 2 (2n + 1) (2n + 3) {( + 1) ( + 2) - 



It is evident that this curve will have two rectilinear asymptotes parallel to the 

 axis of y, whose equations are 



"2s 



x = 



n (n 1} 





(n + 1) (7i + 2) 



and that on passing these critical values, with increasing x, the sign of y will change 

 from positive to negative. The curve will pass through the origin, while when x is 

 very large it will approximate to the parabola 



. 

 J ~ 



(n + I) 3 



Hence it must consist of three branches as in the annexed diagram, where the 

 dotted lines represent the rectilinear and parabolic asymptotes. 



