522 



MR. IVORY ON THE EQUILIBRIUM OF A MASS OF 



another. The inadvertence with which the investigation of Clairaut is chargeable, 

 seems not to have been noticed, at least it is not remedied, by any of the authors 

 who have subsequently handled the subject. 



The attraction of the stratum being admitted, its effect becomes a subject for ma- 

 thematical investigation. We may suppose a stratum of homogeneous fluid bounded 

 by two surfaces of any figure, ABC and ah c, and we may 

 estimate the pressure tending to move an interior particle 

 P in any direction, which is caused by the attraction of the 

 stratum upon the contained fluid. Let dm represent an 

 elementary portion of the stratum ; x\ ?/', z\ the coordinates 

 oi dm ; x,y,z those of P ; and/ the distance between P and 



dm. The direct attraction oi dm on P is equal to -y^ ; and the partial attractions 

 tending inward parallel to x, y, %, are respectively 



dm X — a^ dm y — y' dm z — z' 



or, which is the same. 



/ ' P 



f 



dm df dm df dm df 



~P'~dl-' f^'T^' ~r'~d^ 



and the total partial attractions on P of all the matter of the stratum will be 



, pdm , rdm , f*dm 



djj. ijj^ djj^ 



dx ' dy ' dz ' 



the sign of integration extending to all the molecules of the stratum. Now if jo re- 

 present the intensity of pressure, we shall have 



and by integrating, 



dy 



rdm 



dy + 



dz 



dz ; 



/dm. 

 —r is the sum of all the molecules of the stratum divided by 



their respective distances from P, and C is the like sum at any arbitrary point which 

 may be assumed in the inner surface of the stratum at a, and may be joined to P by 



a narrow canal having any direction : and if we write I / -^ J for C, that is, for the 



sum of all the molecules of the stratum divided by their respective distances from a, 



the value of jo in the formula 



pd m r nd m~] 



p=J-t-Ut^ 



will be equal to the intensity of pressure urging the particle P in the direction of the 

 canal. It appears, therefore, that the effect of the attraction of the stratum to move 



