578 Profs. A. Dendy and J. W. Nicholson. Influence of 



Another question of some importance, to which a prehminary statement 

 may now be devoted, is that of curvature. The well known experiments of 

 Chladni showed that the effect of bending a bar, so that it begins to 

 approximate to a tuning fork, is to make the nodes of the vibrations 

 approach the point of maximum curvature in the centre. Thus the two 

 nodes of the fundamental vibration of a free-free bar, originally at a distance 

 of 0"224 from its ends, move inwards towards the centre as the bar is bent. 

 A diagram given by Chladni, showing the rate of approach of these nodes, is 

 reproduced- by Barton.* The spicule with which we are concerned is 

 somewhat bent, and this result is therefore of importance. The bending is, 

 however, so slight that the shift of the nodes due to this agency may be 

 regarded as additive to that produced by inequalities in the cross-section, 

 and it is not necessary to obtain a quantitative solution of the very difficult 

 problem of a bent bar of non-uniform cross-section. 



The lateral vibrations of bars of variable cross-section have not been 

 investigated hitherto, in the strict quantitative sense, as regards the 

 positions of the nodes, in any case ; although Kirchhofff determined the 

 periods of vibration of a bar which was in the form of a thin cone, and of 

 another bar which has no definite relation to the present problem. A 

 mathematical solution of the complete problem for several cases has, 

 however, been obtained by one of us for the purposes of the present investi- 

 gation. Full details of these results are contained in another paper 

 communicated to the Society. They include the case — important from its 

 approximation to that of the spicule treated in detail here — of a bar which 

 consists effectively of two equal thin cones with their flat ends in contact, as 

 in fig. 1. If 21 be the length of such a bar, and q a multiplier, dependent on 



Fig. l. 



the period, it can be shown that the primary mode in which the bar vibrates 

 symmetrically, so that the curvature of the axis at some particular instant is 

 of the form given in fig. 2, is determined by the equation J 3 [2^/ (ql)] = 0, 

 and the nodes are at a distance x from either free end determined by 

 J 2 [2^/ (qx)] = 0. The J's represent Bessel functions, and the roots of these 

 equations are already known from their occurrence in other physical 



* ' Text Book on Sound,' p. 296. 



t ' Berlin Monatsber.,' 1879, pp. 815-828. 



