232 Proceedings of the Royal Irish Academy. 



These equations become, in the case of an incompressible fluid — 

 dp cPx 



^ - F - ^ (Z\ 



dy~ ^ dt^' ^^^ 



d% df 



where f is the common pressure of the fluid (equal in all directions) 

 at any point {x, y, z). 



It will be noted, that the last three equations of (2), depending on 

 couples, disappear; because, in consequence of the mobility of the 

 particles of the fluid, mter se, internal couples or twists become 

 impossible. 



The Laplacian equations of motion, in polar co-ordinates, are 

 usually deduced from (3), by transformation of the co-ordinates, from 

 X, y, S-, referred to fixed axes, where the axis of x is the axis of rota- 

 tion ; that of y an axis perpendicular to x, and fixed in space ; and 

 that of z, an axis perpendicular to those of x, y ; to r, 6', (f)', where r 

 is the radius vector, 0' is the north polar distance, and ^' is the 

 angular distance from the plane of x, y, of the meridian of any 

 moving particle. 



Instead of referring the forces to fixed co-ordinates, I refer them 

 to the following moveable rectangular co-ordinates : — 



Axis of x'. 



Let H denote the sum of the forces at any point, acting along the 

 radius vector [neyative towards the centre, and 2}ositive from it.) 



Axis of y'. 



Let S denote the sum of the forces at any point acting in the 

 meridional moving plane, and perpendicular to r {positive towards the 

 equator, and negative towards the pole.) 



Axis of z . 



Let T denote the sum of the forces at any point acting perpendicu- 

 larly to the two former directions, or in the direction of the tangent 

 to the small circle of latitude {negative against the rotation, and 

 positive with it.) 



Let r, 6', ^', denote the polar co-ordinates in their most general 

 form. The alteration in pressure produced by a change in r is similar 

 to that produced by a change in x, y, z, of the first three of equa- 

 tions (2) (because they are all linear magnitudes), and denotes a. force 

 acting to or from the centre ; but the alteration in pressure produced 

 by a change in angular direction by a change in 0' or ^' is no longer 



