234 M. C. Sondhauss on the Sounds 



rence requires that at least one equation of condition shall be 

 satisfied, as is shown in my previous paper. 



On the other hand, when an equation of the second order be- 

 tween x, y, z is derivable from an integral equation involving even 

 a single arbitrary function, it is clear from what has herein 

 appeared that the given equation may be replaced by a pair of 

 equations between x, y, z,p } q and their differential coefficients of 

 the first order, which pair will be derivable from a pair of inte- 

 gral equations between x,y,z,p,q involving at least one arbitrary 

 function. The occurrence of this arbitrary function will, as has 

 been seen, split each of the equations to be satisfied, viz. (5a) , 

 into two, thus giving us four in all, while we have but three 

 disposable quantities, viz. E, f, u y wherewith to satisfy them — 

 a circumstance from which it follows inevitably that the occur- 

 rence of integrals of the kind treated of in this paper must be 

 conditional. 



At the same time, nothing has herein appeared which shows, 

 nor am I aware of any argument which tends to show, that a par- 

 tial differential equation of the second order between x, y, z will 

 not always be satisfied by a single integral equation free from 

 arbitrary functions. 



6 New Square, Lincoln's Inn, 

 February 22, 1868. 



XXV. On the Sounds produced by a Jet of Water. 

 By C Sondhauss*. 



IN some posthumous papers of F. Savart there is an investiga- 

 tion of the sounds which are formed when water issues from 

 short tubes placed in vessels. The sounds were most easily 

 produced when the height of such tubes or apertures was equal 

 to their diameters; and Savart found in this case that the 

 number of vibrations of the notes which belong to the same 

 scale was as the square roots of the pressures of water, and 

 inversely as the heights of the tubes. Where the apertures had 

 other dimensions, owing to the difficulty of determining the note 

 no more could be said than that, with equal pressures, the 

 ratio of the numbers of vibrations did not seem greatly to differ 

 from the inverse ratio of the lengths of the tubes. 



When the water issued into air, a note was only obtained in 

 case the height of the aperture was between half and double the 

 diameter ; when the water flowed into water, a tone resulted if 

 the height of the aperture amounted to -^ the diameter. 



Almost simultaneously M. Sondhauss made an extended in- 



* From an abstract in the Fortschritte der Physik, vol. xxi. by Dr. 

 Rober, of the original paper which formed part of the " Programme der 

 Real-Schule zu Neisse," PoggendorfFs Annalen, vol. cxxiy, p. 1. 



