a Paper by Dr. Sondhauss. 213 



n= ^0988P" • (IXl) 



"=5^91498? (C) 



The rational formula (C) was first given by Helmholtz in m*3 

 admirable paper in Crelle, on Vibrations in open Pipes ; it is 

 only strictly applicable to openings of circular form. The dif- 

 ference between (A) and (VII.) is never very great, being on one 

 side when L is small, and on the other when L is large, and ac- 

 cordingly vanishing for some intermediate value. The greatest 

 difference is shown in (VIII.) and (B) when L is very large. I 

 therefore consider Dr. Sondhauss' s opinion and anticipation to 

 be in the main justified by my investigation, when he says, " I 

 remark that I regard the formula (VII.) .... not merely as an 

 empirical formula useful for interpolation, but am convinced 

 that it forms the theoretical expression of a natural law. From 

 the zeal with which the field of mathematical physics is now cul- 

 tivated, we may expect that the laws which I have discovered 

 experimentally will soon be proved by analysis." But I must 

 observe that (A) is only true subject to a series of limitations, 

 which Dr. Sondhauss seems scarcely, if at all, to have contem- 

 plated. All the dimensions of the vessel (with a partial excep- 

 tion of the length of the neck) must be small compared with the 

 quarter wave-length, and the diameter of the neck must be 

 small against the linear dimension of the body of the vessel. 

 The latter condition excludes the case of S small or nothing, to 

 which Dr. Sondhauss pushes the application of his formula. 

 But there is a rational formula proper for closed cylindrical tubes, 

 as has been proved by Helmholtz in his paper on open pipes, to 

 which Dr. Sondhauss refers, but apparently without availing 

 himself of the results. It runs, 



n= % , " (D) 



but is only strictly true when &* is small against L. Although 

 I am of opinion that (A) and (D) and the transition between 

 them cannot be comprehended in the same theoretical investiga- 

 tion, yet it is easy to adjust (A) so as algebraically to include (D). 

 Thus 



a g~* 



"EF / - ■ w • • W 



VF : MMR 



becomes, when S (which now refers to the volume of the body 

 only) is put equal to zero, 



