542 MR. A. E. H. LOVE ON THK SMALL FREE VI1WATIONS 



This gives two types of motion. 



In the first, the motion is partly tangential and partly radial ; the expressions for the 

 displacements are c 



u = 2 P, cos-- v/ 



i=l C 



v = 0, 



(103) 



<r a-iri ^ . iirz , 



' j 



n 



where the equation for K is c 2 = iV, or 



(1 - <r) - 2 _ .., 2 , , 



- - 



and p 2 a 2 /> = w* 2 , i being any integer. 



The displacement is, for each normal type of vibration, wholly tangential along 

 the circles sin iirz/c = 0, and wholly radial along the circles cos iirz/c = ; there are 

 no points or lines of no displacement. The frequency depends on the length and 

 radius of the shell, and the ratios of the intervals for consecutive tones depends on cr, 

 i.e., on the material of the shell. 



In the motions of the second type the displacement is purely tangential, and is 

 expressed by 



10 =0, 

 where the equation for the frequency is 



> = ' /' 4- 1 - 



v= 2 Q,-8in- e'f*, }. ...... (105) 



i = If 9 



J 



or 



4/>, 2 = (2i+ I) 2 7r 2 n/c 2 p ........ (106) 



In this case the circles sin (2i + t) 7r2/2c = 0, are nodal lines. The frequency 

 varies inversely as the length of the cylinder, and the intervals between consecutive 

 tones are independent of the material of the shell. 



Note. July, 1888. In the paper as read some examples were next given of the 

 application of the method to problems of equilibrium. These are now withdrawn, as 

 of little physical interest, and not directly relevant to the subject of the paper (see 

 Summary). 



