438 Mr. Ivory on the Theory of the Figure of the PlaJiets. 



surface, and all the other properties that occur in the analysis 

 of Laplace, are strictly true, because they may all be demon- 

 strated by the ordinary rules. If this should'occasion any 

 difficulty, it is easily removed. In place of the element of the 

 surface of the sphere d s, substitute its value or ^ vj/ sin vj/ c? ^ ; 

 then, observing that the arcs \J; and <p are independent of one 

 another, the second of the formulae (2) may be thus written, 



/I 



(r- — a j + — —yj = 



and the integral y(y — y) d <p, taken between the limits (p = 

 and ^ = 2 TT, will be reduced to 2 tt (H'<^' — i/)' The refrac- 

 tory terms have now disappeared ; and the investigation may 

 be completed either by the method proposed by Laplace, or 

 by the ordinary rules. Thus in whatever manner we view this 

 analysis, if we push our reasoning till the clouds of obscurity 

 are completely dispelled, we are uniformly brought to one con- 

 clusion, namely, that it must be restricted to rational and in- 

 tegral functions of three rectangular co-ordinates. 



If, laying aside analytical symbols and operations, we wish to 

 contemplate the real grounds of the method in the nature of 

 the things concerned, we shall find that all the difficulties arise 

 from the author's considering as differentials the portions of 

 the stratum that stand upon the elements of the spherical sur- 

 face, which is allowable only when the small masses of matter 

 are at a great distance from the point of contact. If I wish 

 to estimate the relative attraction of the mountain Schehallien 

 on a point at a great distance on the surface of the ocean, it 

 may be sufficient to divide the mass of the mountain by the 

 square of the distance. But if the attracted point be close to 

 the mountain, as in Dr. Maskelyne's experiment, the former 

 method would be entirely erroneous : we must now divide the 

 mountain itself into differentials, and sum up the attractive 

 forces by the rules of the integral calculus. The equation at 

 the surface of the spheroid is therefore liable to limitations, 

 which depend upon the law of the attractive force. 



Most of the foregoing observations are to be found in a 

 paper presented to the Royal Society in 1811*. Since that 

 time there has been no question about the genei'al demonstra- 

 tion in the Mecaniqiie Celeste for an attractive force propor- 

 tional to any power of the distance ; but, in the case of nature, 

 or when the attraction is inversely proportional to the square 

 of the distance, some attempts have been made to prove the 

 equation at the surface of the spheroid without specifying 

 particularly the function that expresses the thickness of the 



* Phil. Trans, for 1812. 



stratum. 



