406 Dr. H. Bateman on some 



on this curve we shall have one pair o£ values of X and Yt 

 consequently there will be just one corresponding point Q 

 in the plane II. As P describes the curve C, the point Q 

 will generally describe a curve Y. Let us consider the case 

 when both and T are small closed curves. 



The particular type of electromagnetic field which will 

 now be discussed is of such a nature that the flux of force 

 across the closed curve C is equal to Z d(~K, Y), where 

 d(X, Y) denotes the area of the curve V and Z is a uniform 

 function of x, y, z. and t. If (Z, X, Y) are regarded as the 

 rectangular coordinates of a point M in a space S, the flux 

 is represented by the volume of a cylinder. 



Now consider a small closed volume in the x, y, z space, 

 and let a time be associated with each point as before; then, 

 by considering the flux across the boundary of the volume, 

 we see that the electric charge within the volume is repre- 

 sented by an element of volume d(X, Y, Z) in the space S. 

 Expressing the quantities Z d(X, Y) and d(X, Y, Z) in terms 

 of the variables w, y, z, t, we may write * 



Z d(X, Y) = E r %, z) + E y d(z, x) 4 E z d(x, y) -cH^(.r, *) 



-cRyd{y,t)-cR z d{z,t) y 



d[X, Y, Z) = pd(x, y, z) - pv z d(y, z, t) —pv y d{z, x, t) 



-pv z d(x,y,t). 



The transformation is easily effected by writing t 



d(,j > e) = 1 1, i \ d(w > *> z) = 



etc., 



where dx, B.v, "dx, etc. are independent sets of. increments of 

 the variables x, y, z, t. We easily find that 



_ r7 B(X,Y) _ 1 ;/ 5(X, Y) i 



dx, 



dy, 



dz 



Bx, 



By, 



Bz 



~dx. 



d/A 



-bz 



B(X,Y,Z^ 3(X,Y,Z: 



V • a: 



/3'z = — 



y,*)' rz " B(y,*.0 



J 



* The integral forms were introduced into electromagnetic theory by 

 Mr. R Hare-reaves, Camb. Phil. Trans. (1908). See also H. Bateman, 

 Proc. London Math. Soc. ser. 2, vol. viii. (1910). The coefficients 

 E x , cll x , etc. are generally uniform functions of x, y, z, and t. 



t This is equivalent to assuming that the element of area is a small 

 parallelogram and the element of volume a small parallelepiped. 



