Elect Ac Waves through Tubes, 127 



dx/ds, dy/ds being the cosines of the angles which the tangent 

 (ds) at any point of the section makes with the axes of 

 x and y. 



Equations (2) and (7) are met with in various two-dimen- 

 sional problems of mathematical physics. They are the 

 equations which determine the free transverse vibrations of a 

 stretched membrane whose fixed boundary coincides with 

 that of the section of the cylinder. The quantity P is limited 

 to certain definite values. Is*, k<?, . . ., and to each of these 

 corresponds a certain normal function. In this way the 

 possible forms of R are determined. A value of R which is 

 zero throughout is also possible. 



With respect to P and Q we may write 



r=2+f> • • • • • < 9 > 



Q=f -f; .■-,'.■ m 



where <f and i)r are certain functions, of which the former is 

 given by 



dV dQ dU 



There are thus two distinct classes of solutions ; the first 

 dependent upon <f>, in which R has a finite value, while 

 •^ = ; the second dependent upon fy, in which R and <f> 

 vanish. 



For a vibration of the first class we have 



P=*cty/da 9 Q=d(/>/dy, .... (12) 



and (V 2 + P)(£ = (13) 



Accordingly by (11) 



<*>=$*, (14) 



and v _imdR n _im dR . _. 



k 2 ~dx' ^tftfp ' * * {lb) 



by which P and Q are expressed in terms of R supposed 

 already known. 



The boundary condition (7) is satisfied by the value ascribed 

 to R, and the same value suffices also to secure the fulfilment 

 of (8), inasmuch as 



pAe ,/v dy _ im dR 

 ds ds ~ k 2 ds ~" 



L2 



