Vibration of a System in its Gravest Mode. 569 



in which n=0, 1, 2, &c. This form satisfies Laplace's 

 equation and the condition of symmetry since cosines of 6 

 alone occur. When 6 = \it, it reduces to (6). It remains 

 only to secure the reduction to (7) when r=c, and this can 

 be effected by Fourier's methods. It is required that from 



0=0 to e=w 



2A 2 „ +1 cos(2^ + l)<9=-7 (l+ cos20)+£ 4 (l- cos40)-. . . 



... (9) 

 It will be convenient to write 



A„* 1 =£A 1 « 1 + y 4 A J » 1 + ..., . . (10) 

 so that 



SA^\cos(2n+l)0=(--l)*--cos2s0. . . (11). 



In (11) s may have the values 1, 2, 3, &c. 



The values of the constants in (11) are to be found as usual. 

 Since 



2 j '"cos {2n + 1) 6 . cos (2m + 1)6 d6 



vanishes when m and n are different^ and when m and n 

 coincide has the value \ir, and since 



2 2 "{(-l)*-cos2s0}cos(^-fl)0d<9 



Jo 



=(-i)-+- 



we get 



. m = /_ lv+ »2 r _ l _J ^_ 1 



A 2n+1 V ; ^ | 2n + 2*+l 2n + l 2n-2s+lj" 



• • • (12) 



in which s=l, 2, 3, &c, n = 0, 1, 2, &c. 



The value of -^ in (8) is now completely determined when 

 q 2 , &c. are known. The velocity-potential (/> is deducible by 

 merely writing sines, in place of cosines, of the multiples of 6. 



We have now to calculate the kinetic energy T of the 

 motion thus expressed, supposing for brevity that the density 

 is unity. We have in general 



2T =j^> < 13 > 



where dn is drawn normally outwards and the integration 

 extends over the whole contour. In the present case, how- 

 ever, d<p/dn vanishes over the circular boundary, so that the 



2n+2s+l 2n + l 2n — 2s + 1 



