1889.] Vibrations of an infinitely long Cylindrical Shell. 447 



4>mn 1 /OA x .* 



or _p2 = _ -- ................. (30),* 



m-\-n arp 



as may be seen from (22), by putting s = 0. 



On the other hand, the vibrations of (29) are nearly purely axial. 

 In terms of m and w 



_ 



/> m 



Now, if ^ denote Young's modulus, 







so that p 2 = ., ..(33). 



f 



This is the ordinary formula for the longitudinal vibrations of a rod, 

 the fact that the section is here a thin annulus not influencing the 

 result to this order of approximation. 



Another extreme case worthy of notice occurs when s is very great. 

 Equation (24) then reduces to 



...... (34); 



so that & 2 becomes a function of u and s only through (a, 2 -}-s 2 ), as 

 might have been expected from the theory of plane plates. The first 

 root relates to flexural vibrations ; the second to vibrations of 

 shearing, as in (18) ; the third to vibrations involving extension of 

 the middle surface, analogous to those in (22). 



It is scarcely necessary to add, in conclusion, that the most general 

 deformation of the middle surface can be expressed by means of a 

 series of such as are periodic with respect to z and 0, so that the 

 problem considered is really the most general small motion of an 

 infinite cylindrical shell. 



[Another particular case worth notice arises when 5 = 1, so that 

 (24) assumes the form 



+ 4/i 2 (& 2 -^ 2 )(2M4-l) = ..... ... (35). 



As we have already seen, if yttbe zero, one of the values of & 2 vanishes. 

 If /i be small, the corresponding value of & 2 is of the order /* 4 . Equa- 

 tion (35) gives in this case 



This equation is given, in a slightly different form, by Love (loc. cit., p. 523). 



