220 Prof. Cballis on Hijdrodijnamics. 



Thus we have a differential equation of two variables, by the in- 

 tegration of which T may be found. The value of this func- 

 tion being ascertained, we can determine, first, the amount of 

 that portion of the impressed velocity which, not being accom- 

 panied by condensation, at least so far as the first approxima- 

 tion indicates, takes effect as if the fluid were incompressible, 

 and then the amount which, being accompanied by the conden- 

 sation which the law of propagation requires, is propagated 

 together with the condensation into space. [I have given in- 

 stances of the determination of these two velocities for given 

 disturbances in the Transactions of the Cambridge Philosophical 

 Society, vol. vii. part iii. pp. 310-346.] The motion and 

 condensation of all the elements not immediately impressed are 

 to be inferred from the laws governing the mutual action of 

 the parts of the fluid, which were demonstrated independently 

 of arbitrary conditions. From this argument it follows that 

 the condensation propagated from a centre does not vary in- 

 versely as the distance, as is inferred from hydrodynami- 

 cal principles which I have proved to be defective, but in- 

 versely as the square of the distance. (To this point I shall 

 have to refer again.) Also the argument by which it was 

 shown that a solitary wave, either of condensation or rarefaction, 

 may be propagated without undergoing change when the motion 

 is perpendicular to a plane, equally applies to the condensation 

 accompanying central propagated motion. 



If, as a second example of central motion, we suppose a 

 portion of the fluid, contained within certain limits of dis- 

 tance from the centre, to be condensed at a given instant in an 

 arbitrary manner without initial velocity, it would be found on 

 the same principles that two equal waves of condensation would 

 be generated, and that one would be propagated towards, and 

 the other from, the centre, the condensation and velocity in both 

 varying inversely as the square of the distance. I consider that 

 the mode of solving this problem exhibited towards the end of 

 my paper on the principles of hydrodynamics, in the Philoso- 

 phical Magazine for February 1853, is in accordance with the 

 principles I am now explaining, and needs no correction. 



It will probably be noticed that no difficulty like that which 

 the foregoing argument is intended to meet, presented itself in 

 the problem of the central motion of an incompressible fluid. 

 The explanation of this difference may be readily given. For 

 an incompressible fluid the equation (e) becomes 



dr \r r 1 / 



Now by the general law of rectilinear motion, which is deduced 



