in reply to Professor Challis. Sl-Q 



Prof. Challis's argument, as it now stands, may be divided 

 into the following heads : — 



1. The condensation s may be expressed by r~^¥{r—at), 



2. The function F may be given arbitrarily from r—at^=-h 

 to r—at^=.h-\-\ and may be supposed to vanish beyond those 

 limits, so that the wave of condensation is comprised between 

 the spheres whose radii are at-\-h and a/ + Z> + A. 



3. The fluid may be supposed at rest beyond the outer, and 

 within the inner boundary of the wave of condensation. 



4. Therefore we may introduce two rigid envelopes, &c. 

 The third head I wholly deny, and have already, as I con- 

 ceive, disproved. (Phil. Mag., vol. xxxiv. p. 54.) I have 



shown that unless / F(r)</r=0, the fluid cannot be at rest 



both outside and inside the wave of condensation. 



It is vain to reply (as at p. 91) that the expression for the 

 velocity is not proved. It is proved on the hypothesis (I do 

 not myself regard it as an hypothesis) of spherical waves, as 

 Prof. Challis does not seem to deny. It is plainly illogical 

 reasoning to make the hypothesis of spherical waves, obtain a 

 part of the results to which the analysis leads on that hypo- 

 thesis, refuse to attend to other results to which the analysis 

 equally leads, and then, oti the gratuitous assumption of the 

 possibility (according to the hypothesis under trial) of a state 

 of things which does not result from the partial solution already 

 obtained, and which the analysis, if wholly carried out, would 

 have proved to be impossible, to argue that the hypothesis is 

 absurd. 



In conclusion, I will merely explain what I meant by saying 

 that I did not regard the hypothesis of spherical waves, as an 

 hypothesis. I consider it axiomatic that the initial conden- 

 sation and initial velocity may be conceived to be given arbi- 

 trarily in a mass of elastic fluid ; at least if no abrupt varia- 

 tions be supposed to exist in the initial condensation or velo- 

 city: such abrupt variations, in so far as they are admissible, 

 may be afterwards considered as limiting cases of continuous 

 variations. Consequently we may without absurdity conceive 

 the initial condensation and initial velocity to be arranged 

 symmetrically about a centre. But in this case the conden- 

 sation and velocity must be symmetrical with respect to the 

 centre during the whole motion ; because if we draw any two 

 radii vectores from the centre, whatever we can say of the one 

 we can say of the other ; and therefore the velocity will be 

 directed to or from the centre, and the condensation and ve- 

 locity will be functions of /• and of /. This will be equally 

 true whether we neglect or take account of the development 



