and Spontaneous Diffusion of Sound and Lialit. 251 



occurs, that the absurdity should be accounted for and got 

 rid of. I have adverted above to two modes in which a dis- 

 turbance at A may be rectilinearly transmitted from A, so as 

 to have effect at any point P. In the first it was stated that, 

 as matter of experience, the effect takes place independently 

 of any particular conditions under which the fluid was dis- 

 turbed at A ; in the other the fluid is assumed to be disturbed 

 in a particular manner, and under certain conditions. The 

 former of these modes may be called, for distinction, free or 

 spontaneous motion, and the other constrained or arbitrary 

 motion. Now, in the received analytical theory of hydro- 

 dynamics, the determination of the rate of propagation is made 

 to depend on the conditions of the arbitrary mode of dis- 

 turbance, whereas, as matter of fact, the propagation and rate 

 of propagation are both independent of the particular form of 

 disturbance. This contradiction of the theory by experimental 

 facts sufficiently accounts for the reductio ad absurdum, 

 which, consequently, signifies that the theoretical reasoning 

 is at fault and requires to be rectified. To show how this is 

 to be done is the purpose of the following investigation. 



The problem to be solved is to account for the fact of a 

 rectilinear transmission of effect from a disturbance made at 

 a certain point A to any point P in the surrounding fluid, 

 without reference to any assigned mode of disturbing the 

 fluid. Since there are no arbitrary initial conditions of the 

 motion, it follows that the kind of motion must be indicated 

 by some genei'al analytical circumstance. It will here be 

 supposed that the required analytical circumstance is that the 

 differential function udoc + vdy + wdz is an exact differential 

 in all cases. On this supposition it may be assumed that there 

 will be a rectilinear axis of motion between A and P, the 

 motion relative to the axis satisfying the condition of the 

 integrability of udx + vdy + ivdz. Also in order that there 

 may be motion persistently along the axis, it is necessary that 

 the condensation pertaining to the motion should be symme- 

 trically disposed about the axis, so that both the condensation 

 and consequent transverse velocity will be functions of the 

 distance (r) from the axis. Consequently, supposing that 

 the axis of z coincides with the given axis of motion, we have 

 to assume that, with reference to that axis, 



(d ./</>) = udx + vdy + wdz } 



f being a function of r, and (f> a function of z and /. Ac- 

 cordingly, 



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