409 



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 witli 

 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 



