322 Mr. Airy in reply to Prof. Challis 



confine myself to another mode of investigation, of which the 

 results will be beyond doubt, namely, that by rectangular co- 

 ordinates. 



The point in dispute may be well stated in ProfessorChallis's 

 words : " I have certainly considered it possible that the velo- 

 city of the fluid at a given distance from the centre to or from 

 which it is directed, may, at a given instant, be different in 

 different directions from the centi-e, provided there be no 

 abrupt variation." 



Now I undertake to show that, 



1st. If the motion of the particles be directed to or from a 

 centre (or approximately so, the whole investigation being 

 approximate), the motions must be the same in all directions, 

 in phase as well as in coefficient. 



2nd. If, in order to support Professor Challis's expression 

 for the movement of the particles to or from a centre, we 

 suppose other movements perpendicular to these radii to 

 coexist, then it will be shown that the possibility of such a 

 combination of movements is not proved, and that (as it de- 

 pends upon finding two quantities which shall satisfy three 

 equations) the presumption is, that it is impossible. 



I proceed now with the investigation. 



If u, V, IV, be the velocities in the directions x,y, z, of a 

 particle which at the time t has the coordinates x, y, z (those 

 velocities being expressed in tei'ms of x,y,z, and t), and if 

 g be the density at that point and p the corresponding press- 

 ure, and if g = k p ; the equations to the motion of that 

 particle (no accelerating force being supposed to act) are the 

 following : 



