Prof. Challis on the Hydrodynamical Theory of Vibrations. 299 



(art. 34) all in the same phase, it will follow that beyond a cer- 

 tain finite distance from the cylindrical surface the sum of the 

 positive condensations may be as nearly as we please equal to 

 the sum of the negative condensations. Thus the resultant con- 

 densation will vanish, and there will be neither transverse nor 

 longitudinal motion. As that distance must be many multiples 

 of X, it can be a small quantity only in case X be extremely small. 

 The magnitudes of the longitudinal vibrations increase from zero 

 at the limiting distance till they acquire a maximum and uniform 

 value at a certain distance within the cylindrical surface ; and the 

 transverse vibrations, increasing from zero at the exterior limit 

 till they reach a maximum near the cylindrical surface, after- 

 wards diminish till they disappear at the interior limit by the 

 counteraction of opposite vibrations. The thickness of the 

 cylindrical shell which, within its interior and exterior surfaces, 

 includes the whole of the transverse motion, will be less as the 

 breadth of the waves is less ; and if the waves be of extremely 

 small breadth, it is conceivable that that space and the interior 

 cylindrical space occupied by the motion which is exclusively 

 longitudinal, may together only form a cylinder of very small 

 radius. Thus motion included within such a cylinder would 

 be propagated to an unlimited distance without lateral divergence. 



44. It might be objected that, although this reasoning may hold 

 good so long as the component motions are such that the con- 

 densations are symmetrically arranged about the axes, it does 

 not apply to components which have undergone the modifica- 

 tions described in articles 40 and 41. To this objection it may 

 be replied that the resolutions of the motion and condensation 

 there considered are confined, as is shown in art. 42, to distances 

 from the axis very small compared to X, whereas the transverse 

 motion extends to distances which are many multiples of X. 

 When symmetrical motion and condensation about an axis is 

 separated by any cause into two parts, each part will have posi- 

 tions of no condensation at distances from the axis the same as 

 those at which the condensation was zero previous to the sepa- 

 ration. There is, in fact, no reason for concluding that, at posi- 

 tions not extremely close to the axis, the separation produces 

 any other effect than a partition of condensation between the 

 two parts, the directions of the motion remaining the same. 

 Thus the asymmetry near the axes, within very small distances 

 from them, merges into symmetry. This remark may suffice to 

 meet the above-mentioned objection. 



45. There remains for consideration another point, viz. the 

 determination of the laws of the propagation of undulations. 

 As these laws are not arbitrary, they must admit of being ascer- 

 tained independently of the supposition of any case of arbitrary 



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