104 Mr Bryan, On the beats in the [Nov. 24, 



As we shall require to apply the variational equation of energy, 

 we must calculate 8?& the virtual work of the effective forces of 

 the system. Now if f lf f % denote the transversal and radial 

 accelerations of any point, the well-known formulas applicable to 

 polar co-ordinates give 



J1 a+w dt ( ' ' 



= v + 2oow, 

 f 2 = iv — (a + w)(co+ vfa) 2 

 = — aco 2 + w — 2cov — woo 2 

 (neglecting squares and products of the small displacements v, w). 

 Hence 



8® = fVi^ +/M) aade 

 Jo 



= aa \ {— aw 2 8w + (v + 2cow) 8v 

 Jo 



+ (w-2cov-wco 2 )8w}8d (2). 



The variation of potential energy will consist of three terms 

 representing respectively the work done against the tension T in 

 stretching the circumference, the work done against the attracting 

 force, and that of bending the cylinder. We shall denote the 

 potential energies due to these three causes by W v W 2 and V 

 respectively, and their variations will be 8W 1} BW 2 and 8V. 



To find 8W i , let e be the extension produced by the displace- 

 ments (v, w) in the arc add. By writing down the stretched length 

 of the arc we have 

 (1 + ef (add) 2 = dw 2 + (a + w) 2 (d6 + dv/a) 2 ; 



. 2/ dv\ 1/ dv\* 1 (fdw\ 2 a dv 



and therefore, to the second order 



1 / dv\ 

 ' = a{ W + Td) 



1 {fdw\ 2 _, dv) ,_. 



Hence 



BW^TTSe.ade 



Jo 



by integrating by parts. 



