411. 



enlarged point of view ; and D'Alembert has extended his researches 

 to other figures beside the elliptic spheroid, and has invented a method 

 of investigating the attractive force of a body of any proposed figure, 

 and composed of strata, varying in density according to any given 

 law ; but his method, though ingenious, is destitute of the requisite 

 simplicity. 



Laplace has also treated this extremely difficult question with his 

 usual skill, and has deduced the relation between the radius of the 

 spheroid and the series for the attractive force, upon a point without 

 or within the surface, in a manner admirably simple when the com- 

 plicated nature of the question is considered. 



In the course of his investigation, Laplace lays down a theorem, 

 which he affirms is true at the surfaces of all spheroids that differ 

 but little from spheres. This proposition is enunciated in the Me- 

 canique Celeste in the most general manner, comprehending every 

 case in which the attractive force is proportional to any power of 

 the distance between the attracting particles. But the demonstration 

 which Laplace has given of this proposition appears to Mr. Ivory not 

 to be conclusive. It is, says he, defective and erroneous, because a 

 part of the analytical expression is omitted without examination, and 

 is rejected as evanescent in all cases ; whereas it is so only in par- 

 ticular spheroids, the radii of which are expressed by rational and 

 integral functions of a point in the surface of a sphere ; and though 

 the quantities which Laplace has omitted are then really equal to 

 nothing, yet, says Mr. Ivory, this does not happen for any reason 

 assigned by Laplace, but for a reason that has no manner of con- 

 nexion with anything touched upon in his demonstration. 



In order to avoid all discussions which are not of real use to the 

 inquiry into the figures of the planets, Mr. Ivory confines his atten- 

 tion chiefly to the case of nature, in which attraction follows the law 

 of the inverse proportion of the squares of the distances. But he does 

 also briefly examine the theorem of Laplace, in the general sense in 

 which it is laid down in the Mecanique Celeste ; and he admits, that 

 when the exponent of the law of attraction is positive, and not les\ 

 than unity, then the demonstration of Laplace is not liable to so much 

 objection, and the theorem is in that case true to the full extent of 

 his enunciation ; but he observes, that when the exponent is nega- 

 tive, then certain quantities become infinitely great, instead of being 

 equal to nothing, as the theorem of Laplace would require them to 

 be. 



The writings of no author on any subject, says Mr. Ivory, are en- 

 titled to more respect than those of Laplace on the subject of phy- 

 sical astronomy ; and, consequently, it was not till after the most 

 mature reflection that he has ventured to dissent from an authority 

 for which he has the utmost deference. But in a work of so great 

 extent as the Mecanique Celeste, which treats of so great variety of 

 subjects, all very difficult and abstruse, it could hardly lie expected 

 that no slips or inadvertencies have been admitted, even by an author 

 whose knowledge of the subject he treats is so profound, and the 



