paper, it would be wrong to omit to state that the developments which it contains, on the investigation of the attraction in the simpler case, are highly ingenious, and exhibit a perfect command of analysis.
The second subject is the criticism upon the method used by La- place in the third book of the ' Mecanique Celeste,' for the compu- tation of the attraction of spheroids of any form differing little from spheres, and the substitution of a method purely analytical for some of Laplace's operations which are founded on a geometrical consi- deration. The papers which contain Mr. Ivory's remarks on these subjects are two papers and an appendix in the volume for 1812, and one in that for 1822. The remarks on Laplace's theory ad- verted to two points. One of these was the faultiness of his reason- ing as relates to the evanescence of the attraction of the particles included between the spheroidal and a spherical surface when the attracted particle was brought very near to the surface. The other was a limitation of the generality of Laplace's assumption for the form of the function expressing the distance between the sphere and the spheroid, to a rational function of the coordinates of each point. With regard to the first of these subjects, it seems impossible to deny that Laplace had, in the greater part of his investigation, left the interpretation of his suppositions in some obscurity ; and Mr. Ivory has, with remarkable acuteness and analytical skill, exposed the de- fects of Laplace's investigation on his interpretation of the suppo- sitions. Yet we must observe that the limitation expressed by La- place (" supposons de plus que la sphere touche le spheroide, &c.") appears to be entirely overlooked by Mr. Ivory, and that this limi- tation, when its effects are fairly examined, completely removes the objection. As to the second subject, if is, we believe, allowed by Mr. Ivory himself, that there is no failure in the investigation if the function for the distance between the sphere and the spheroid, though not explicitly rational, admits of being expanded in a converging series whose terms are rational ; the only case undoubtedly that can ever occur in physical application. The analytical process which Mr. Ivory substituted for a part of Laplace's is extremely beautifid.
To show the estimation in which Mr. Ivory's talents and labours were held by Laplace himself, we may here quote a remark from Sir Humphry Davy's Address in 1826, on the award of the Royal Medal to Mr. Ivory. " I cannot pretend," says our, then, distin- guished President, " to give any idea of the mathematical resources displayed in the problems, and which even the most accomplished geometer cotild not render intelligible by words alone ; but I can speak of the testimony given by M. de Laplace himself in their fa- vour. That illustrious person, in a conversation which I had with him some time ago on Mr. Ivory's first four communications, spoke in the highest terms of the manner in which he had treated his sub- ject; one, he said, of the greatest delicacy and difficulty, requiring no ordinary share of profound mathematical knowledge, and no common degree of industry and sagacity in the application of it."
The investigations to which we have just alluded are those upon
