Mr. A. Schuster's Electrical Notes, 11 



in (4), but is arbitrary. If, on the other hand, F'jG', H' are 



not solutions of (1), they cannot be converted into solutions 

 by the terms involving %. 



That F, G, H, as given in (5), do not necessarily satisfy (1) 

 may be shown as follows. Writing F,, G l7 B^ instead of 

 F, G, H on the left-hand side of (5), we obtain from these 

 equations, 



1 ax 



and with the help of (3) and (4), 



4™=g-V 2 F 1 . (6) 



Comparing this with 



^=£^F, (2) 



it is not correct to draw the conclusion that F x is a solution 

 of (2), for equation (6) still involves on the right-hand side 

 the quantities F, G, H. The correct conclusion is that 



V 2 F 1 = V 2 F (7) 



We may treat equation (5) in a different way, and deduce 

 from them 



dF x dQj d,R, = d¥ f rfGK dW 



dx dy dz dx dy dz 



If, then, F l5 G l5 H : were equal to F, G, H, as Maxwell puts 

 them, it would follow that 



dW dG' dW A /Q . 



dx dy dz 



and this would show that F', G 7 , H / are solutions of (1) 

 without the addition of the terms involving %. 



It may be shown that in all legitimate cases equation (8) is 

 satisfied ; but before proceeding to discuss the conditions on 

 which this depends, 1 shall treat of a special case which will 

 bring out clearly the points to which I wish to draw attention. 



Let u, v, iv be electric currents deducible from a current 

 function (p such that 



_ d(p _ d(f> _ d(j> 



"~dv> r ~dy> tn ~dz~ ; 



and let cb= — . 



The lines of flow in this case are identical in shape with the 



