FIFTH REPORT —1835. 
4 
The calculation in these cases was far from easy. The same 
entanglement occurred here which mathematicians had already 
found so perplexing in the problem of the Figure of the Earth ; 
namely, that the attractions could not be calculated without 
knowing the form of the mass, and yet the form depended on 
the equilibrium of those very attractions. The only ways in 
which this difficulty could be surmounted were, either to devise 
methods for finding the attractions of bodies which should be 
applicable to all forms; or else, to assume some form, and to 
calculate the attraction, and then to modify the assumption so 
as to make it approach to a fulfilment of the conditions of equi¬ 
librium. The first-mentioned method was in the course of in¬ 
vention by Legendre and Laplace at the time of Coulomb’s 
researches. Some of the earliest memoirs in which it is used 
appear in the very same volumes of the Transactions of the 
French Academy as Coulomb’s memoirs on electricity and mag¬ 
netism (1782—1789). But it required a long period to fami¬ 
liarize even the best mathematicians with this method, and its 
application to electricity was reserved for a later period, and for 
the skill of M. Poisson. In the mean time Coulomb applied, 
with great industry and ingenuity, such artifices as were obvious 
to a geometer of that time. For example*, in treating the case 
of two spheres, in order to determine the proportion in which the 
electric fluid distributes itself between them when one is elec¬ 
trised and brought into contact with the other, he supposed, as 
an approximation, the fluid to be uniformly spread over the sur¬ 
face of each, in order to calculate its attraction ; although it is 
manifest that, in fact, the mutual repulsion of the parts of the 
fluid will make its density vanish at the point of contact, and 
increase gradually up to the point diametrically opposite. In 
order to correct this process, he supposes a small segment of the 
sphere surrounding the point of contact to be void of fluid, and 
finds the effect of this segment as if it were a circular plane. 
In a case in which a sphere is in contact with other spheres at 
two opposite points or poles , he finds the attraction of its fluid 
on two suppositions ; one, that it is spread uniformly, another, 
that it is all collected in a ring at the equator of the sphere : the 
real distribution will be intermediate between these two supposi¬ 
tions, because the density of the fluid at the poles vanishes, and 
increases gradually up to the equator; hence Coulomb takes the 
mean of the results of the two suppositions, as more approximate 
to the truth than either of them. He pursues methods of this 
kind with a very unsparing expenditure of labour; for instance, 
* Acad. Paris. 1787. 
