2 Proceedings of the Royal Irhh Academy. 



the point r in the plane of u. Proofs of these theorems will be found in 

 the Proceedings of the London Mathematical Society, series 2, vol. i., parts 

 2 and 3. If we pass on to moving distributions of electricity, we have a scalar 

 potential \p = j de V^(p~\ and a vector potential 



{F,G,II) =j{f/{r), Mr), Mr))deVf\ 



and the vectors {X, Y, Z) and («, /3, 7) are related to them just as above. 



For points outside the electrical matter, it is possible by simple differen- 

 tiation to show that ip, F, G, H, X, Y, Z, a, j3, 7, all satisfy the equation 

 V^ - V'^d' I dt~ = 0, and that the following relations hold : 



d^ dF dG dS -f^ 



dt dx ay dz 



dX dY dZ ^ 



dx ay az 



V I {X r, Z) = curl (a, i3, 7), 

 ct 



- li («> 13, 7) = curl {X, Y, Z), 



^ + ^+1^=0. 



a>j dy dz 



It is the object of this paper to find out what the above relations become 

 when the point in question is inside the electrical matter. The relations to 

 be obtained will differ from the above in much the same way as the equation 

 of Poisson V'* U + 4:Trp = differs from that of Laplace V^ Z7 = 0, where 

 U is the ordinary attraction potential. It will appear that the new relations 

 wall be exactly the equations of Maxwell as amended by FitzGerald. 



The method adopted in the ordinary attraction theory affords a hint as to 

 how we must proceed. Let a sphere be drawn so as to include the point 

 {x, y, z), and of such a radius that the density may be considered uniform 

 throughout its volume. It is then obviously necessary to consider only the 

 electricity inside this sphere, which we will suppose to move with the distri- 

 bution. We begin with the simplest case of a sphere moving without 

 rotation, the centre of which is at the point x-^it), yi{t), Zi(t) ; and the radius 

 is equal to a, the coordinates of any point inside being Xo + Xi{t), 2/0 + 2/i(0' 

 «o + ^i(Oj ^^cl p being the volume density. We shall write 



''-'0 + 1/0" + ^0 = ^0 J 



so that To :j> a, and {x - x^ {a))- + {y - y^ (u)- + {z - Zi {u)y = r (?/)- ; 



