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



where the term \ s p* is derived from the terms uBu which represent 

 the virtual work of the effective forces due to the longitudinal 

 components (u) of the displacement, and it is important to notice 

 that \ can never be negative. Although this last statement is not 

 obvious from the variational equation required for the treatment 

 of a revolving shell, it becomes evident from the consideration that 

 in a non-revolving shell the variational equation leads to the same 

 equation for p as the principle of conservation of energy, and that 

 in the equation of energy the term containing X s p 2 will arise from 

 the kinetic energy of the longitudinal motion (u). This kinetic 

 energy is of course essentially positive; or, in other words, the 

 whole kinetic energy of the system is greater than it would be if 

 the longitudinal motion were neglected. Hence \ is positive. 

 The nodal angular velocity, 



and the number of beats per revolution. 



s*+\ -1 



2s 



s* + \ + 1' 



are therefore both greater than they would be if there were no 

 longitudinal motion. 



The limiting case is that of a plane circular plate revolving 

 about an axis perpendicular to its plane. Here v, w are both zero, 

 and the nodal radii are fixed relatively to the revolving mass, the 

 vibrations being unaffected by the rotation. The nodal angular 

 velocity is therefore a>, and the number of beats per revolution 

 is 2s. 



Now in a communication read before the British Association 

 at Leeds, I announced that experiments with two different 

 champagne glasses attached to a microscopist's turn-table gave 

 about 2'6 and 2'2 beats per revolution respectively for the gravest 

 tone. While there is nothing contradictory in the former result, 

 the latter is too small to be compatible with our theory. As the 

 numbers were found by counting the beats during about eight 

 revolutions of the table and the mean of 26 observations was 

 taken, it is impossible that the discrepancy can arise wholly from 

 errors of observation. A further possible source of error was the 

 want of uniformity in the angular velocity of the glass. As a 

 matter of fact, however, the beats seemed, if anything, most rapid 

 when the glass was first set in motion, and as it was not brought to 

 rest again during the interval in which the beats were counted, I 

 rather doubt this as* the cause of the difference. On the whole I 

 should be rather inclined to favour the idea that the discrepancy 



