492 MR. IVORY ON THE EQUILIBRIUM OF A MASS OF 



theory of Clairaut is therefore perfect, and it possesses all the elegance which might 

 be expected from the talents of the author. On the other hand, if the forces in action 

 depend upon the very figure to be found, as must always happen when the particles 

 attract one another, the equation of the surface will not be explicitly known, because 

 the differential coefficients are derived, in part at least, from the unknown figure of 

 the fluid. Since quantities which depend entirely upon what is sought are not elimi- 

 nated from the final equation, the ordinary rules of mathematical investigation would 

 lead us to infer, either that the problem is not solved, or that it is indeterminate, and 

 admits of many solutions. It is allowed on all hands that there is a mutual connexion 

 between the figure of a mass of fluid and the attractions it exerts upon its particles : 

 the relation which these two things, alike unknown, must bear to one another in the 

 case of equilibrium, is expressed by the equations of the upper surface and of tlie 

 interior level surfaces ; and therefore it seems hardly possible to deny that these equa- 

 tions are indeterminate. What is wanting to complete the solution of the problem 

 cannot possibly be supplied by any abstract or mathematical properties which the 

 indeterminate equations may possess ; and hence arises a suspicion that there is an 

 imperfection of the theory, proceeding, probably, from some necessary condition 

 having been overlooked. 



None of the observations that have been made go the length of charging with in- 

 accuracy any of the properties of Clairaut's theory, or any of the equations which 

 express those properties. An equilibrium of a mass of fluid entirely at liberty cannot 

 exist, unless all the conditions of that theory be fulfilled. The question is, whether 

 tliose conditions be sufficient to determine completely the figure of equilibrium in all 

 hypotheses respecting the forces. It is no small imperfection that the principal points 

 of this theory have not been deduced from the nature of an equilibrium in a manner 

 independent of opinion or arbitrary assumptions. If a strict mode of investigation 

 had been followed, we should have been in possession of a just criterion for ascertain- 

 ing in any particular case, whether all the conditions required for an equiHbrium 

 were fulfilled or not. But in solving problems of this kind, it is often thougiit suffi- 

 cient to prove some enumerated properties, or to obtain certain algebraic equations, 

 which unavoidably introduces obscurity and occasions a want of evidence ; since it 

 can hardly be supposed that the same properties, or the same equations, will bear 

 alike upon a gi'eat variety of problems diffijring from one another in the nature of the 

 forces urging the particles of a fluid. 



Is not the principle, that the equilibrium of a mass of fluid is in all cases secured 

 when every individual particle is pressed equally by all the canals issuing from it 

 and terminating in the surface, an opinion or an assumption ? That the property is 

 general, no one will doubt. But when the fluid consists of attracting particles, the 

 forces urging the particles and the pressures of the canals both vary when the upper 

 surface of the fluid is made to change : and may it not be alleged that the variation 

 of the figure of the mass may be such that the pressures of all the canals may still 



