432 Mr. Ivory on the Theoi-y of the Figure of the Planets 



tance to distinguish between the foundations of our reasoning 

 and the mathematical processes, of whicli the office is not to 

 supply, or stand in the room of, principles, but to deduce the 

 consequences that flow from them. In the present instance, 

 although we admit the first approximation to be true, yet, un- 

 less the grounds of the method are fully cleared up, we must 

 remain doubtful with regard to the succeeding steps in which 

 we attempt, by reiterated operations, to approximate more and 

 more to the exact figure of equilibrium. 



At the present time, when so much has been done, and is 

 still doing, to determine the figure of the earth experimentally, 

 it seems proper likewise to reconsider the theorj'. With re- 

 gard to the writings of Maclaurin and Clairaut, no examina- 

 tion is required. If these authors have taken a less extensive 

 view of the problem, the grounds and the manner of their 

 reasoning need no elucidation. But it appears from what has 

 been said, that the method delivered in the Mecaniqiic Celeste 

 for the equilibrium of spheroids little different from spheres, 

 is not entirely free from objection in its physical principles ; 

 and the illustrious author has occasioned some difficulties by 

 the manner in which he has laid down the fundamental proofs 

 of the analysis he employs. I propose therefore to make 

 some observations, first on the analysis, and secondly on the 

 physical principles of the theory of the figure of the planets 

 contained in the third book of the Mecanique Celeste. In very 

 intricate cases it appears to be the destined lot of humanity to 

 approach the truth very slowly, and by reiterated efforts. Dif- 

 ficulties arise and are overcome in succession ; and this pro- 

 gressive improvement is, perhaps, no where more strongly ex- 

 emplified than in the branch of philosophy to which our at- 

 tention is at present directed. The theory on which I pro- 

 pose to remark has been before the public for more than a 

 quarter of a century; it has great merit, and has been highly 

 applauded; and if we now presume to make it the subject of 

 examination there can be but one apology, which is, the just- 

 ness and the truth of the strictui'es to be made. 



I. Conceive a spheroid very little different from a sphere; and 

 having assumed any point upon its surface, let a straight line 

 r be drawn inwards, at right angles to the surface, so as to 

 reach very near the middle or centre. From the extremity 

 of ?• describe a sphere having for its radius aline a nearly 

 equal to, but less than, r ; then it is plain that any radius of 

 the spheroid may be represented by the expression a (1 + «?/'), 

 « being a small coefficient, of which the square is to be neg- 

 lected, and y a function of two variable arcs that determine 

 the position of the radius. We shall denote the arcs here 



mentioned 



