138 Prof. Challis on the Mathematical Theory oj 



that the free motion of an clastic fluid was parallel rectilinear 

 motion, because from the nature of the case that kind of motion 

 must be unique and definite in its character. 



18. It may, however, be urged per contra, that parallel recti- 

 linear motion may certainly be impressed on the fluid under 

 arbitrary circumstances, and that the investigation of the rela- 

 tion of the velocity to the density would, in a particular case of 

 disturbance, lead to the very equations adduced above, which 

 therefore must admit of interpretation relative to fluid motion. 

 To this argument I reply that the investigation of the laws of 

 the mutual action of the parts of the fluid must precede the con- 

 sideration of particular cases of motion, and that, by taking this 

 course, it will be found that the above equations are not appli- 

 cable to any instance of parallel rectilinear motion arbitrarily 

 impressed on the fluid. To this point I shall recur in a sub- 

 sequent stage of the reasoning. 



19. Again, udx + vdy + wdz would be integrable per se if the 

 motion were in straight lines drawn from a centre. It will spare 

 the necessity of any lengthened consideration of this case, to say 

 that if the equations derived from the hypothesis of this kind of 

 rectilinear motion involved no contradiction, neither would there 

 be contradiction in the equations just considered, because pa- 

 rallel rectilinear motion may be regarded as central rectilinear 

 motion at an infinite distance from the centre. We must con- 

 clude that as the one hypothesis led to contradictions, so would 

 the other, and consequently that the kind of rectilinear motion 

 resulting from the mutual action of the parts of the fluid is not 

 central rectilinear motion. 



20. There remains to be made the hypothesis of an axis of 

 rectilinear motion ; that is, of motion in a straight line coexist- 

 ing with curvilinear motion. That this is a legitimate hypo- 

 thesis will appear from the consideration that, as the law of 

 rectilinearity was not an inference from a combination of all the 

 general equations, we are not under the necessity of supposing 

 the motion to be wholly rectilinear. Now, since there are no 

 arbitrary conditions to take into account, we are at liberty to 

 suppose the axis of rectilinear motion to be one of the axes of 

 coordinates, for instance the axis of z ; and we have then to 

 express analytically that the motion parallel to the plane xy 

 vanishes for points on this axis, and that, for points immediately 

 contiguous to it, udx -f vdy + wdz is an exact differential. These 

 conditions may be satisfied as follows. Let /be a function of x 

 and y, and </> a function of z and t, and, since the above expres- 

 sion is to be an exact differential, assume that 



[d ,f(j)) = udx + vdy + wdz. 



