182 ANNUAL OF SCIENTIFIC DISCOVERY. 



of the other end of the clam, and then the vibrations of the earth cease. 

 Standing in front of the dam, and placing a pole on the bed of the river, 

 directly under the fall/the pole was violently agitated, although there are two 

 feet of back-water through which the water must pass before it reaches the 

 pole. In the falling sheet, the wavelets are concavo-convex, and not double 

 convex, that is, the internal surface corresponding with an external convex- 

 ity is concave. 



Prof. W. B. Rogers remarked, that the wave-like divisions of the descend- 

 ing sheet of water were probably referable to the same general law which 

 has been shown by Savart and Plateau to obtain in the case of a stream 

 flowing from an apeiture in the bottom or side of a vessel. These philoso- 

 phers have proved that, at a certain distance from the point of discharge, the 

 stream, although seemingly continuous, is in reality divided into separate 

 parts; and Plateau explains this subdivision by the preponderance of trans- 

 verse cohesive forces in the column, when its length exceeds its thickness by 

 more than a determinate amount. In this case the sides of the stream are 

 drawn together at intervals, and the mass is thus broken up in separate sec- 

 tions, which, by further cohesive action, are moulded into drops. Thus every 

 such stream, at some distance below the aperture, loses its straight outline, 

 and assumes the form of a series of enlargements and contractions, which, 

 at a still greater distance, become visible as a succession of drops. 



It is interesting to remark that a regular system of vibrations occurs in the 

 streams thus affected by Avave-like subdivisions, enabling them to produce 

 strong musical tones, when suffered to strike upon an elastic surface, and to 

 communicate similar musical vibrations even to the reservoir itself. In this 

 action we discern someAvhat analogous conditions to those of powerful vibra- 

 tory movements attending the descent of large masses of water over dams. 



ON THE MATHEMATICAL THEORY OF SOUND. 



In a paper on the above subject read before the British Association, 18-58, 

 by the Rev. S. Eanishaw, the author adverted to the circumstance, that the 

 only impediment to the complete development of the mathematical theory of 

 sound has hitherto been the difficulty of integrating the partial differential 



/tfw\2 dill dZil 



equation f \ y~7-=/ t yr, As an approximative mode of surmounting this 



difficulty, it has been usual to assume (-j-\= 1. But the author suggested 



that the legitimacy of that step is by no means evident; and that the true 

 test of the allowableness of it is a knowledge of the change, which must 



d2y d^y 



take place in the constitution of the atmosphere, in order that = p. 



dtz dx'2 



may be the exact equation of motion. In this way it will be seen whether 

 the physical change, represented by assuming ( - - Y= 1, be of such a mi- 

 nute character as to be allowable. But the aiithor stated that he had found 

 the requisite change in the constitution of the atmosphere must be such that 

 it must increase in A'olumc with an increase of pressure, a constitution the 

 very opposite of that of the natural atmosphere. From this it was inferred 

 that the equation which represents the properties of sound does not admit of 



f-dii \2 



the assumption \~--J 1- The reason Avhy this assumption, though ana- 



