314 Mr. Challis on the Theory of the 



must have travelled from their origin by very irregular paths. 

 The intensity of the sound is proportional to m. 



The velocity of propagation along any path whatever, is 

 equal to a, because neither reflection nor the action of the parts 

 of the fluid on one another alters this velocity. 



From the preceding considerations it will not be difficult to 

 .perceive, that the motion along the line of propagation from 

 one point to another, may be the same as if the fluid were 

 contained in a tube, the transverse section of which is every 

 where indefinitely small, but difterent at difterent points, and 

 the axis of which coincides with the line of propagation. The 

 position of the axis, and the law, either continuous or not, ac- 

 cording to which the transverse section varies, will be known 

 from the data of the problem to be solved, coupled M'ith the 

 laws of spherical propagation demonstrated in Arts. 15 and 16. 

 We shall have 



V = as = mF {(7 — at) (e), 



where (t may be measured along the axis from a fixed point in 

 it, instead of being taken of arbitrary length, along different 

 straight lines drawn in the direction of propagation, to all of 

 which the axis is a tangent. The form of F is given by the 

 initial disturbance, and m is inversely proportional to the radius 

 of the tube, supposing the transverse sections to be circles. If 

 the sections be every where the same, m is constant, and v = as- 

 F((7 - at), as in straight cylindrical tubes. 



The equations (e) apply to the motion which takes place 

 along the axis of any straight musical pipe of finite length, 

 even when prolonged beyond its mouth into the exterior fluid. 

 For this line must be a line of propagation, as no reason exists 

 why there should be deviation from it in one direction rather 

 than another. It would be difficult to ascertain the alteration 



