ON FARADAY'S LINES OF FORCE. 209 



With respect to the history of the present theory, I may state that the 

 recognition of certain mathematical functions as expressing the "electro-tonic 

 state " of Faraday, and the use of them in determining electro-dynamic 

 potentials and electro-motive forces is, as far as I am aware, original ; but the 

 distinct conception of the possibility of the mathematical expressions arose in 

 my mind from the perusal of Prof. W. Thomson's papers " On a Mechanical 

 Representation of Electric, Magnetic and Galvanic Forces," Cambridge and 

 Dublin Mathematical Journal, January, 1847, and his "Mathematical Theory of 

 Magnetism," Philosophical Transactions, Part I. 1851, Art. 78, &c. As an 

 instance of the help which may be derived from other physical investigations, 

 I may state that after I had investigated the Theorems of this paper 

 Professor Stokes pointed out to me the use which he had made of similar 

 expressions in his "Dynamical Theory of Diffraction," Section 1, Cambridge 

 Transactions, Vol. ix. Part 1. Whether the theory of these functions, consi- 

 dered with reference to electricity, may lead to new mathematical ideas to be 

 employed in physical research, remains to be seen. I propose in the rest of 

 this paper to discuss a few electrical and magnetic problems with reference to 

 spheres. These are intended merely as concrete examples of the methods of 

 which the theory has been given ; I reserve the detailed investigation of cases 

 chosen with special reference to experiment till I have the means of testing 

 their results. 



EXAMPLES. 

 I. Theory of Electrical Images. 



The method of Electrical Images, due to Prof. W. Thomson*, by which 

 the theory of spherical conductors has been reduced to great geometrical sim- 

 plicity, becomes even more simple when we see its connexion with the methods 

 of this paper. We have seen that the pressure at any point in a uniform 

 medium, due to a spherical shell (radius = a) giving out fluid at the rate of 



a* 

 4*JV units in unit of tune, is kP outside the shell, and kPa inside it, 



where r is the distance of the point from the centre of the shell. 



* See a series of papers "On the Mathematical Theory of Electricity," in the Cambridge and 

 Dublin Math. Jour., beginning March, 1848. 



VOL I. 27 



