536 MR. A. E. H. LOVE ON THE SMALL FREE VIBRATIONS 



which can be satisfied either by 



B^O, and l+(l + o-)(a+l)P.-(l-o-),i --"=<>, (80) 



rt/4 



or by 



tJP 

 A = 0, and y8(/3+ l)P,-2/ = ... (81) 



This gives two types of motion. 



In the first the motion is partly tangential and partly radial. Since P a (/*) cannot 

 have equal roots, u and w cannot vanish together, or there are no lines of no displace- 

 ment. The displacement is purely radial along the lines dP (p.)/dp, = 0, and 

 purely tangential along the lines P a (p.) = 0. The ratios of the frequencies of the 

 component vibrations of this type depend on cr, i.e., on the material of the shell. 



In the second type the motion is purely tangential, every point moving through a 

 distance along the parallel to the edge through it, which is the same at all points of 

 the parallel. The lines dP ft (p.)/dp. = are nodal. The ratios of the frequencies of 

 the component vibrations are independent of the material of the shell. 



20. For a hemispherical bowl /u, = at the edge. 



(1.) In the motions of the first type P a (p-) is to vanish with p, ; hence, a. is an odd 

 integer, or, i being any integer, we have 



(2 + * 2 )/(l + c) = (2i + 1) (2t + 2) = 01 say, 

 where 



c [K - 4 (1 + cr)/(l - cr)] =(! + <r)/(l - cr). 

 This gives 



* 4 (1 -cr)-2K 2 (l +3cr + ft))+ 4 (co - 2) (1 -f cr) = ; . . ' . (82) 



this equation has always real roots. 



If K; 2 , K? be the roots, and p it p\ the corresponding values of p, according to the 

 formula p z a 2 p = n/c 2 , then 



(83) 



- /* 2 ) {P 2( - 

 o a f l 



17= 0, 



f~V(l -^)(; 



L 



, = o 

 To get arithmetical results, let us choose cr = ^ ; the equation for K 2 becomes 



+ 2)-2} =0, 

 and KJ, K'J are given by the table : 



