274 Mr. Challis on the Theori/ of the 



of which differ by an insensible quantity, as this line will satisfy 

 the same differential equations, it indicates a possible motion, 

 which is consequently of that bizarre and irregular kind, which 

 Lagrange first demonstrated to be the general character of the 

 vibrations. The particular form, however, of this line is given, 

 when the particular mode of the disturbance which caused the 

 motion is given. I have endeavoured to exhibit as clearly as 

 possible, the mechanical reasons of this kind of motion. 



In the next place, the bearing of the theory on the musical 

 sounds produced in tubes, is briefly considered, and particular 

 attention is paid to the mode in which the air vibrates in a tube 

 open at both ends, because on this point, the view I have 

 taken, leads to an inference which is at variance with the re- 

 ceived theory. 



The equation which gives the motion in space of two dimen- 

 sions is integrated approximately, and the approximation is shewn 

 to be such, that the integral will ajjply with accuracy to almost 

 all cases that can occur. Euler's integral of the equation that 

 applies to the motion in space of three dimensions, which has 

 ever since his time been considered to be particular, is here 

 .shewn to be the proper general solution, and adequate to solve 

 all the cases of small motions. This view of it is justified by 

 its ajiplication to some problems of interest, particularly to 

 oblique reflections, and the problem of resonances. In conclu- 

 sion, I have stated as a result of the whole i^receding investiga- 

 tion, the manner in which analysis points out the laws of any 

 phenomena, the theoretical enquiry into which conducts to the 

 solution of a partial differential equation. 



I. Motion in Space of one Dimension. 

 2. To begin with the simplest case, let us suppose a portion 

 of the medium to be inclosed in a very slender cylindrical tube 



