Royal Society. 145 
nearly equivalent to this by Laplace, but his process was trouble- 
some; that by Mr. Ivory is remarkably simple and elegant. Al- 
though this transformation constitutes the most valuable part of the 
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 ‘ Mécanique Céleste,’ 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 iis interpretation of the suppo- 
sitions. Yet we must observe that the limitation expressed by La- 
place (“‘supposons de plus que la sphére touche le sphéroide, &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, it 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 beautiful. 
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 could not render intelligible by words alone; but I can 
speak of the testimony given by M. de Laplace himself in their fa- 
your. ‘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 
Phil. Mag. S. 3. Vol. 22, No. 143, Feb, 1843. L 
