Vibrations upon the Form of Certain Sponge- Spicules. 579 



problems.* Taking the first root in each case as corresponding to the 

 fundamental vibration of symmetrical type, we find 



= 6-379, 2y/(qx) = 5135, 



■and therefore x/l = - 648, 



•or, the distance of the node from the free end bears to the whole length of 

 the rod the ratio 0'324, as against - 224 for a uniform rod. 



Each of the unsynmietrical vibrations has one node at the centre of the 

 Tod. This is evident without analysis. When such a structure vibrates, 

 therefore, the nodes which should be prominent, as corresponding to the 

 more fundamental and therefore stronger vibrations, are three in number — 

 one at the centre and one at - 324 of the length measured from each end. 



The uniform rod and double cone are limiting cases. When the rod is not 

 oent, the nodes for any intermediate configuration can be approximately 

 predicted, but must be between the limits 0"224 and 0'324. When the rod 

 is sharply conical the second value should be nearly reached. The corre- 

 sponding value for a nearly double-parabolic rod, as in fig. 3, is 0'29. Bending 

 of the axis of any rod must always make these theoretical values higher by 

 pulling the nodes in towards the centre. 



Fig. 3. 



Quoting a more general result of the mathematical investigation above, 

 we may state that, if a bar is composed of two equal halves, each consisting 

 of a portion of the solid formed by rotating the curve y = Ax n about the axis 

 of x — in practice the axis of the spicule — the distanco of the symmetrical 

 uode from the free end is the product of the length and a numerical factor 

 0-074 + 0-300 (4n + l)/(4w + 2) 



with great accuracy. 



* Lord Rayleigli, ' Theory of Sound,' vol. 1, p. 330. 



