322 Mr. Ivory on the Figure of Equilibrium of a Fluid, 



surface, is invariably the same over all the surface of A. 

 Now the supposed equilibrium of A will not be disturbed by 

 pressures of the same intensity exerted at all the points of its 

 surface; and therefore a new body of fluid in equilibrium, 

 namely, A + SA, is obtained. Continuing to reason in like 

 manner, the original mass A may be enlarged to any dimen- 

 sions by the addition of successive strata, at the same time 

 that an equilibrium is preserved at every step. It is easy to 

 make the procedure independent of the fii'st supposition, that 

 A is in equilibrium ; for as the reasoning holds whether A is 

 great or small, we may suppose it so small that any forces 

 inherent in its own particles and tending to change its figure, 

 are overpowered and annihilated by the accumulated pressure 

 of the incumbent strata. 



When the property of equilibrium, demonstrated in these 

 few words, is put in equations as Clairaut has done, these 

 equations are found to be the very same as those deduced 

 from Euler's theory. Thus, in point of application, the two 

 methods are entirely equivalent : if one is capable of solving 

 a problem, the other may be used with equal success. 



The investigation of Clairaut is clear and definite. It evi- 



1 • 



dently assumes that there is no cause tending to disturb the 

 equilibrium of A, except the action of the forces at the sur- 

 face of A upon the matter of 8 A. On this account his me- 

 thod fails when there is a mutual attraction between the mass 

 A and the stratum S A. If the mass A attract the matter of 

 the stratum 8 A and cause it to press, it follows necessarily 

 that the matter of 8 A will react, and, by its attraction, will 

 urge the particles of A to move from their places. In this 

 case therefore the equilibrium of A is disturbed by a cause 

 which Clairaut has not attended to ; and unless the effect of 

 this new force is counteracted, the body of fluid A + S A, will 

 not be in equilibrium. The principle of the method suggests 

 a remedy for this omission ; for it is easy to prove that the 

 equilibrium of A will not be disturbed by the attraction of the 

 stratum 8 A, if the resultant of that attraction upon every 

 particle in the surface of A, be directed perpendicularly to 

 that surface, And thus we arrive at the same two inde- 

 pendent conditions for the equilibrium of a fluid consisting of 

 attracting particles, which have been found necessary in every 

 other way of solving the same problem when nothing essen- 

 tial is neglected. (Vide this Journal for August last, p. 81, 

 and for October, p. 274.). 



The principle of Clairaut's method, when enunciated ge- 

 nerally, lies in this, that the supposed equilibrium of A is not 

 to be disturbed by the addition of a stratum ; and therefore 



