PROPAGATION OF ELECTROSTATIC FORCE. 257 



If we write Curl for the differentiating vector operator which 

 Maxwell calls by that name, equations (8) may be put in the form 



47r$) = Curl Curl (iu), 

 whence dto/dt = (4nr) ~ l Curl Curl (i du/dt). 



From dl&jdt we may calculate the magnetic induction 33 by an 

 operation which is the inverse of (47r)~ 1 Curl. We have therefore 



33 = Curl (i du/dt), 



or $=[r-*F'(t-cr) + cr- 2 F"(t-cr)](yk-zj). 



The magnetic induction is therefore zero except in the waves. 



Equations (4) and (9) give the value of dtyt/dt as function of t and 

 T. By integration, we may find the value of 31, Maxwell's "vector 

 potential." This will be of the form of the second member of (4) 

 multiplied by c~ 2 , if we should give each F one accent less, and for 

 an unaccented F should write F 1 , to denote the primitive of F which 

 vanishes for the argument oc . 



That which seems most worthy of notice is that although simul- 

 taneously with the discharge of the system (A, B) the values of what 

 we call the electric potential, the electrodynamic force of induction, 

 and the " vector potential," are changed throughout all space, this does 

 not appear connected with any physical change outside of the waves, 

 which advance with the velocity of light. 



If we now suppose that there is a second pair of charged spheres 

 (c, d), as in the original problem, the discharge of this pair will 

 evidently occur when the relaxation of electrical displacement reaches 

 it. The time between the discharges is, therefore, by Maxwell's 

 theory, the time required for light to pass from one pair to the other. 



It may also be interesting to observe that in the axis of x, on both 

 sides of the origin, xp = r^i, and equation (4) reduces to 



Here, therefore, the oscillations are normal to the wave-surfaces. 

 This might seem to imply that plane waves of normal oscillations 

 may be propagated, since we are accustomed to regard a part of an 

 infinite sphere as equivalent to a part of an infinite plane. Of course, 

 such a result would be contrary to Maxwell's theory. The paradox 

 is explained if we consider that the parts of the wave-motion, 

 expressed by F and F', diminish more rapidly than those expressed by 

 F", so that it is unsafe to take the displacements in the axis of x as 

 approximately representing those at a moderate distance from it. In 

 fact, if we consider the displacements not merely in the axis of x, but 

 within a cylinder about that axis, and follow the waves to an infinite 

 distance from the origin, we find no approximation to what is usually 



meant by plane waves with normal oscillations. 



J. WILLARD GIBBS. 

 New Haven, Conn., March 12 [1896]. 



G. II. R 



