502 Sir Gr. Grre3nhill on the 



but there is this important difference, that whereas e~ x 

 never vanishes for any finite value of x, the function G(x) 

 has an infinite number of positive roots, corresponding to 

 the roots of J o = 0. 



The nth derivative of. C(» is Clifford's (— l) n C n (x), and 

 the nth integral from to x is G_»(#) = x n C n (x) ; so that 

 the roots interlace of G n and O n +i = 0. 



Thence the differential equations 



< ^S^=W, (3) 



i[^ +1 T] + ^ c »w= ' • • • w 



-"g + (» + l)'§ + a = 0; ... (5) 



and, in terms of the Bessel Function, 



J„(2 s/x) = a-i-CLC*) = ^ W C„(#) = **■( - 1)* ^^ ; (6) 

 including J (2 V#) = G(a?), as above. 



2. Take the case of the uniform chain, hanging vertically 

 and vibrating slightly, investigated on the first page of 

 ' Bessel Functions ' by Gray and Mathews, and again 

 at the end of r,he book, in the more general case where 

 the density of the chain is supposed to vary as some nth 

 power of x, the height above the lower end. 



To realise the experiment it is easier to revolve the chain 

 bv hand, bodily in steady motion, and to investigate the 

 permanent shape. The plane vibration will then be shown 

 in the shadow of the revolving chain thrown on the wall. 



Taking the condition of relative equilibrium of the 

 length x above the lowest point, 



*Mt: •^= ' (1) 



with T = o\r as at re^t, a the line density; and putting 

 g = a>H, where I is the height of the equivalent conical 

 pendulum revolving at the same rate co, radians/sec, then 

 differentiating, 



sv*5) + 7 =0 - *^ + ^ + t =0 ' and y = bt (i} 



... (2) 

 as in (5) § 1 with ?i = ; and then I is the subtangent at the 

 lowest point, supposed free. 



