524 Sir G. Greenbill on the 



form (2) except in KirchhofFs case of p—q=l,n=2, the 

 conical rod. 



In the general identification of (A). (B), and (C), begin 

 with h = k'=r, nk= — l 3 s = r + 2k='dr : then the D.E. 



£(- +3 S)=^ w 



has the solution 



y = «-<!_,(»') = z-+io L <y) (9) 



If the rod is of revolution, r + 2 = 2(3r — 2), »'=|, and the 

 solution is obtained of the case given by J. W. Nicholson, 

 Proc. Roy. Soc. 1917, p. 51*. 



Try h=0, &=-l, w=-7- ; then 



£( rr+2 S) = ^ ?/ = C_,(,-); . . (10J 

 and if of revolution, r+2 = 2r— 8, r=10, as on p. 5L9. 



17. If the chain in § 3, whirled round in a plane about a 

 fixed point, is replaced by a flexible rod of cross section K 

 , and moment of inertia K/c 2 , the equations for the lateral 

 vibration when the centrifugal force is taken into account 

 (as for instance in the blade of an air-screw) become, for 

 tension X, shearing force Y, and bending moment B 7 



j + Kp- x = 0, X = ihp — - — , . . . (1) 

 ax y I K y 



(BKpg) . . (3) 



: /B__ d i 

 dx dx 



with g=lco 2 y _i/ = _^; and with E=p?, so that c is the 



^7/ = y 

 gdt 2 X 



hydrostatic head of the substance, density p, for the pressure E, 

 dividing out Kp, and integrating, with \ydx = u, 



J0 d 4 u , , 9 „ d' 2 u u _ . iN 



a D.E. homogeneous in the lengths a, c, k, 1. X. #, contrasted 

 with the undimensional, equation (8) § 1G, and combining 

 the two qualities of the independent lateral vibration of the 

 elastic rod when a> — 0, £=oo, and of the flexible chain of 

 §3, c = ; and apparently intractable in the general case. 



