ON THE STABILITY OF THE MOTION OF SATURN'S KINGS. 305 



d? V d* V d"V 

 The equation _ + _+_ + 4trp = 



is true for any system of matter attracting according to the law of gravitation. 

 If we bear in mind that the expression is identical in form with that which 

 measures the total efflux of fluid from a differential element of volume, where 



-r- , -j , -r- are the rates at which the fluid passes through its sides, we may 



easily form the equation for any other case. Now let the position of a point 

 in space be determined by the co-ordinates r, tf> and z, where z is measured 

 perpendicularly to the plane of the angle $. Then by choosing the directions 

 of the axes x, y, z, so as to coincide with those of the radius vector r, the per- 

 pendicular to it in the plane of <f>, and the normal, we shall have 



dx = dr, dy = rd<f>, dz = dz, 



dV = dV dV^ldV d_V = dV 

 dx~ dr ' dy ~ r d<f>' dz ~~ dz ' 



The quantities of fluid passing through an element of area in each direction are 



dv jj dVl j j dv jj 



-j- rddtdz. -j-r -drdz, j- rd<f>dr, 



dr d<(> r dz 



so that the expression for the whole efflux is 



/oc\ 



I 7 |. ~r 7 . laOji 



r dr d'r r d<f> dz 

 which is necessarily equivalent to the former expression. 



Now at the centre of the ring -ry- may be found by considering the attrac- 



Uw 



tion on a point just above the centre at a distance z, 



dV 



-j- = It 

 dz 



R 



TV = ; , when z = 0. 

 dz 1 a' ' 



1 dV R , 



Also we know -=- = ; , and r = of, 



r dr a* 



i d'V I d'V n R 



so that in any circular ring - + = 2 



an equation satisfied by the former values of L and N. 



VOL. i. 39 



