504 J Sir G. Greenhill on the 



Whirled round in a plane about a fixed point, the D.E. for 



the vibration of the chain is again Legendre's D.E. 



d' 2 x 

 Lowered with constant velocity U, ,v = JJt, ~r^r = 0, 



ar 



d 2 y a _ , d 2 v dv a _ ,_ N 



„ =bcM\=l,c(£), with — =\, (9) 



i= ta ^ = u c -0- - • ■ ( ,0 > 



4. Calling D n (#) or D the associated second solution of 

 the Clifford D.E. (5) § 1, then D n (x) = v^Yn^^x), where 

 Y n is the Neumann function, second solution of BessePs 

 equation. 



Thus D (x) would be required, as well as C (V), in the 

 solution of (2), (4), § 2, if the chain carried a weight W, 

 whirled round at the lower end. 



Eliminating C«, D„ between the two D.E/s of the 

 (5) § 1 form, 



and integrating, 



or ^(Cn+iDn— : C„D„+i) = const,, ... (2) 



the equivalent of (36) p. 16, Gray & Mathews, ' Bessel 

 Functions/ Thence 



d D H A n A „ f d# 



Dn = AOn )^k- • • ( 3) 



5. In the linear differential equation of the second 

 order, with arbitrary variable coefficients functions of :,. 

 the canonical form is written 



Jg+ivo, .... . (1) 



