Lord Rayleigh on Bells. 15 



cylindrical coordinates are z, r, </>, is supposed to undergo 

 such a displacement that its coordinates become 



z -f Bz, r + Br , (/> + B(f>. 



The altered value (ds + dBs) of the element of length traced 

 upon the surface is given by 



(ds + dBs) 2 = (dz + dBz) 2 + (r + Br) 2 {Bcj> + dB<f>) 2 + (dr + dBrf. 



Hence, if the displacement be such that the element is un- 

 extended, 



dz dBz + r 2 d<J3 dS<f> + rBr (d<f>) 2 +drdBr = 0. 

 Now 



7S> dBz j , dBz -,, 

 dbz = -y-dz + -y-. deb, 

 dz dcp T 



dor do?' 

 dBr= -^-dz-¥ rr deb 



dz dcp T 



and by the equation to the surface 



dr dr 



in which, by hypothesis, dr/dcp = 0. Thus 



If the displacement be of such a character that no line 

 traced upon the surface is altered in length, the coefficients 

 of (dz) 2 , (dcp) 2 , dz dcp, in the above equation, must vanish 

 separately, so that 



dS_z drd&r 



dz^dzdz -"' W 



^ + & '=°> (2) 



5^2+£S-o ( 3) 



