_ The methods of investigation were also different. In the 
_ one the force on a given particle of electricity has to be 
determined as the resultant of the attraction of all the 
- other particles, In the other we have to solve a certain 
partial differential equation which expresses a relation 
between the rates of variation of temperature in passing 
along lines drawn in three different directions through a 
point. Thomson, in this paper, points out that these two 
problems, so different, both in their elementary ideas and 
their analytical methods, are mathematically identical, 
and that, by a proper substitution of electrical for ther- 
mal terms in the original statement, any of Fourier’s 
‘wonderful methods of solution may be applied to elect- 
rical problems. The electrician has only to substitute 
an electrified surface for the surface through which heat is 
supplied, and to translate temperature into electric poten- 
tial, and he may at once take possession of all Fourier’s 
solutions of the problem of the uniform flow of heat. 
To render the results obtained in the prosecution of one 
_ branch of inquiry available to the students of another is 
an important service done to science, but it is still more 
important to introduce into a science a new set of ideas, 
belonging, as in this case, to what was, till then, con- 
sidered an entirely unconnected science. This paper of 
Thomson’s, published in February 1842, when he was a 
very young freshman at Cambridge, first introduced into 
mathematical science that idea of electrical action car- 
ried om by means of a continuous medium which, though 
it had been announced by Faraday, and used by him as 
the guiding idea of his researches, had never been appre- 
ciated by other men of science, and was supposed by 
mathematicians to be inconsistent with the laws of elec- 
trical action, as established by Coulomb, and built on by 
Poisson. It was Thomson who pointed out that the ideas 
employed by Faraday under the names of Induction, 
Lines of Force, &c., and implying an action transmitted 
from one part of a medium to another, were not only 
consistent with the results obtained by the mathe- 
maticians, but might be employed in a mathematical 
form so as to lead to new results. One of these new re- 
sults, which was, we have reason to believe, obtained by 
this method, though demonstrated by Thomson by a very 
elegant adaptation of Newton’s method in the theory of 
attraction, is the “ Method of Electrical Images,” leading 
to the “ Method of Electrical Inversion.” 
_ Poisson had already, by means of Laplace’s powerful 
method of spherical harmonies, determined, in the form 
of an infinite series, the distribution of electricity on a 
sphere acted on by an electrified system. No one, how- 
ever, seems to have observed that when the external 
electrified system is reduced to a point, the resultant 
external action is equivalent to that of this point, together 
with an imaginary electrified point within the sphere, 
which Thomson calls the e/ectyic image of the external 
point. 
Now if in an infinite conducting solid heat is flowing out- 
wards uniformly from a very small spherical source, and 
part of this heat is absorbed at another small spherical 
surface, which we may call a sik, while the rest flows 
_ out in all directions through the infinite solid, it is easy, 
by Fourier’s methods, to calculate the stationary tempe- 
4 
rature at any point in the solid, and to draw the iso- 
thermal surfaces. One of these surfaces is a sphere, and 
if, in the electrical problem, this sphere becomes a conduct- 
ing surface in connection with the earth, and the external 
source of heat is transformed into an electrified point, 
the sink will become the zmage of that point, and the 
temperature and flow of heat at any point outside the 
sphere will become the electric potential and resultant 
force. 5 
Thus Thomson obtained the rigorous solution of elec- 
trical problems relating to spheres by the introduction 
of an imaginary electrified system within the sphere. 
But this imaginary system itself next became the 
subject of examination, as the result of the transfor- 
mation of the external electrified system by reciprocal 
vadwi vectores. By this method, called that of electrical 
inversion, the solution of many new problems was ob- 
tained by the transformation of problems already solved. 
A beautiful example of this method is suggested by 
Thomson in a letter to M. Liouville, dated October 8, 
1845, and published in Liouville’s Fournal, for 1845, but 
which does not seem to have been taken up by any mathe- 
matician, till Thomson himself, in a hitherto unpublished 
paper (No, xv.of the book before us), wrote out the 
investigation complete. This, the most remarkable 
problem of electrostatics hitherto solved, relates to the 
distribution of electricity on a segment of spherical sur- 
face, or a dow/, as Thomson calls it, under the influence 
of any electrical forces. The solution includes a very 
important case of a flat circular dish, and of an infinite 
flat screen with{a circular hole cut out of it. 
If, however, the mathematicians were slow in making 
use of the physical method of electric inversion, they 
were more ready to appropriate the geometrical idea of 
inversion by reciprocal radii vectores, which is now well 
known to all geometers, having been, we suppose, dis- 
covered and re-discovered repeatedly, though, unless we 
are mistaken, most of these discoveries are later than 
1845, the date of Thomson’s paper. 
But to return to physical science, we have in No. vii. 
a paper of even earlier date (1843), in which Thomson 
shows how the force acting on an electrified body can be 
exactly accounted for by the diminution of the atmospheric 
pressure on its-electrified surface, this diminution being 
everywhere proportional to the square of the electrifi- 
cation per unit of area. Now this diminution of pressure 
is only another name for that ¢ezsion along the lines of 
electric force, by means of which, in Faraday’s opinion, 
the mutual action between electrified bodies takes place. 
This short paper, therefore, may be regarded as the germ 
of that. course of speculation by which Maxwell has 
gradually developed the mathematical significance of 
Faraday’s idea of the physical action of the lines of force. 
We have dwelt, perhaps at too great length, on these 
youthful contributions to science, in order to show how 
early in his career, Thomson laid a solid foundation for 
his future labours, both in the development of mathe- 
matical theories and in the prosecution of experimental 
research. Mathematicians however will do well to take 
note of the theorem in No. xiii., the applications of which 
to various branches of science will furnish them, if they 
be diligent, both occupation and renown for some time 
to come, 
