8 
FIFTH REPORT— 1835. 
attracted point Only, and the other on the corresponding thick¬ 
ness only ; and this resolution, and the peculiar properties of 
these factors, it is which so much facilitate the treatment of 
attractions. These functions were introduced by Legendre, so 
far as regards their use for figures of revolution, and are thus 
employed in the Savans Strangers , tom. x.* and in the Mem. 
Acad. Par. for 1784, read July 7, 1784, published 1787* In the 
Mem. Acad. Par. for 1782, published 1785, is a memoir by 
Laplace, in which the use of analogous functions is extended to 
figures not of revolution ; and in consequence of this step, and of 
Laplace’s frequent use of these functions, they have, as I have 
said, sometimes been designated by his name. So far, however, as 
they have been applied to the calculations of electrical theory by 
M. Poisson, those only have been used which apply to figures of 
revolution, which, as we have seen, are rather Legendre’s func¬ 
tions than Laplace’s. Mr. Ivory, in the Philosophical Transac¬ 
tions for 1S12, treated their properties in a manner which was at 
once allowed by their French inventors to have given them a 
clearness quite new. Mr. Murphy in a separate workt has 
presented them and their application with great analytical ele¬ 
gance and simplicity. The most important parts of Poisson’s 
application of these functions to the case now under notice may 
be found in part 2 of the article Electricity in the Encyclo- 
p cedi a Metropolitana. 
By the use of these functions M. Poisson was enabled J to 
reduce the distribution of electricity upon two spheres which act 
on each other, to a certain functional equation. In the case 
where the spheres are in contact, he solved this equation by 
means of definite integrals, and thus obtained finite and exact 
formulae for the density of the electricity at all the points of the 
two spheres, and for its whole quantity in each sphere. He had 
thus the means of much more rigorous comparison of Coulomb’s 
experiments with his theory than Coulomb himself had been 
able to make. 
The result of this new comparison wag a confirmation of the 
inference from the old one ; and especially in those cases where 
Coulomb’s methods of calculation had been most inadequate, 
as in the instance of two spheres in contact. The experimental 
and theoretical numbers in fourteen such cases came very near 
each other; the mean error being of the quantities them¬ 
selves if the positive and negative errors be allowed to balance 
* According to Legendre’s own reference, which I have not been able to verify. 
f Elementary Principles of the Theories of Electricity, Heat, and Molecular 
Actions: Parti. Cambridge 1833. 
X Mem. Inst. 1811. 
