MEMBRANES AND PLATES 



149 



ingly given by tan 6 = A l /B l . If all the coefficients of order 

 less than s vanish, we have, for small values of r, 



(20) 



The node has then s branches passing through 0, making equal 

 angles TT/S with one another, their directions being given by 

 tan s0 = AgjBg. This is illustrated in the preceding diagrams ; 

 for instance the cases s = 2, s = 3, s = 4 all occur in Fig. 50. 



Fig. 53. 



According to a general theorem stated in 16 it must be 

 possible by combination of the various normal modes of a 

 membrane in suitable proportions, and with proper relations 

 of phase, to represent the effect of arbitrary initial conditions. 

 We do not enter into this; and the theory of the forced vibrations 

 must also be passed over except fora simple example. 



