Vibrations upon the Form of Certain Sponge- Spicules. 577 



reference, the results which are of interest from the present point of view. 

 These relate, of course, entirely to the positions of the nodes. When the har 

 is set in vibration, the various tones are not excited equally strongly, and 

 the nodes which may be expected to occur are only those associated with 

 the two gravest vibrations — or the more fundamental vibrations. Any 

 higher vibration is of necessity accompanied by graver notes, which cause 

 the nodes of the higher vibration to be in motion, so that their significance 

 as points of rest must be diminished or destroyed. If the length of the bar 

 be taken for convenience as unity, and if it be free at both ends, the most 

 fundamental vibration gives nodes at distances 0"2242 from either end, 

 while the second tone has a node in the middle and two others at distances 

 01521 from either end. The first tone is of much greater importance, and 

 the usual vibration must consist mainly of a superposition of the first and 

 second tones. 



If, on the other hand, the bar is fixed at one end and free at the other, the 

 fixed end must be a node. The second tone gives a node at - 2261 from the 

 free end, and the third gives a node at 0"1321 from the free end, and 

 another in the middle of the bar. It is not necessary to consider, for the 

 particular type of spicule dealt with in the present communication, the case 

 of a bar fixed at both ends. 



When attention is restricted to the most important vibrations, we see that 

 the free-free bar gives nodes at 02242 from each end and one in the centre 

 (which is common to many tones), while the fixed-free bar gives a node at 

 0-2261 from the free end. These distances are practically identical. The 

 fixed end in the latter case is also a node. 



This result is not directly applicable, in the quantitative sense, to the 

 spicule with which this communication is concerned, but, at the same time, 

 valuable qualitative information can be derived at once in general terms. If 

 a bar is thick in the middle and tapers to a point at each end, it may be 

 regarded in two ways. In the first place, if it be looked upon as a free-free 

 bar symmetrical about its centre, the nodes, ordinarily at - 2242 from each 

 end, must move inwards towards the more inert centre, in accordance with a 

 general principle of vibratory motion. The bar may, however, if the centre 

 is very thick in comparison with the ends, and therefore almost entirely free 

 from vibratory motion, be regarded as made up of two fixed-free bars placed 

 together but vibrating independently. There must then be a node at the 

 centre, and two others, one for each bar, at a greater distance than - 2261 

 from the free end. The general conclusion, from both points of view, is that 

 a node should be expected at the centre, and two others, one on either side, 

 whose distance from the corresponding free end cannot be so small as 0'2242. 



