358 Note 2: distribution of attracting ftuid 



NOTE 2, ARTS. 27 AND 282. 

 [Distribution in spheres and ellipsoids.} 



The problem of the distribution, in a sphere or ellipsoid, of a fluid, the 

 particles of which repel each other with a force varying inversely as the n th 

 power of the distance, has been solved by Green*. Green's method is an ex- 

 tremely powerful one, and allows him to take account of the effect of any 

 given system of external forces in altering the distribution. 



If, however, we do not require to consider the effect of external forces, the 

 following method enables us to solve the problem in an elementary manner. 

 It consists in dividing the body into pairs of corresponding elements, and 

 finding the condition that the repulsions of corresponding elements on a given 

 particle shall be equal and opposite. 



(i) Specification of Corresponding Points on a line. 



Let A^A Z be a finite straight line, let P be a given point in the line, and 

 let Q l and Q 2 be corresponding points in the segments A-f and PA% respectively, 

 the condition of correspondence being 



It is easy to see that when Q l coincides with A lt Q 2 coincides with A z , and 

 that as Q l moves from A l to P, Q 2 moves in the opposite direction frOm A t 

 to P, so that when Q l coincides with P, Q 2 also coincides with P. 



Let Qi and Q 2 ' be another pair of corresponding points, then 



i i i i 



Qt'P Af ~ PQ t ' PA 2 



Subtracting (i) from (2) 





 Q,'P Q,P PQ 2 ' PQ 2 ' 



QiQi C/Pi 



QSP . Q,P PQ 2 ' . PQ Z 



If the points Q l and Q v ' are made to approach each other and ultimately to 

 coincide, QjQi ultimately becomes the fluxion of Q, which we may write Qi, 

 and we have 



* "Mathematical Investigations concerning the laws of the equilibrium of fluids 

 analogous to the electric fluid, with other similar researches," Transactions of the 

 Cambridge Philosophical Society, 1833. Read Nov. 12, 1832. See Mr Ferrers' Edition 

 of Green's Papers, p. 119. 



