1915-16.] On Systems of Partial Differential Equations, Etc. 301 
Since the equation of wave-motion is closely associated with the partial 
differential equations of Maxwell’s electromagnetic theory, it is natural to 
inquire whether the partial differential equations 
rot 
H - 1 
H - c dt’ 
div E = 0, 
rot E = - 
div H = 0 
1 0H | 
c ~dt > 
( 2 ) 
possess classes of solutions analogous to those already mentioned. This 
question has already been answered in the affirmative in the case of 
solutions of the last type, for it is known that a complex set of solutions 
can be found in which * 
E=*H = ZF(X, Y), 
where F is a vector function of X and Y : one component can be taken to 
be an arbitrary function of X and Y and the other two are then determined. 
The quantities X, Y, Z are certain functions of x, y, z, and t. 
We shall now show that it is also possible to find classes of complex 
solutions of type 
E=iH = F(X, Y, Z), 
where each component of the vector F satisfies a partial differential 
equation in X, Y, and Z. Let us write s = ict, then E must satisfy the set 
of equations 
rot E 4- = 0, div E = 0 . . . (3) 
These equations will be satisfied in consequence of the equations 
rot E = 0, div E = 0 . . . . (4) 
which may be regarded as the fundamental equations of electrostatics in 
the variables X, Y, Z, if these quantities depend on x, y, z, and s in such 
a way that 
0X_0Y_0Z 0Y az dZ 0X 0_X qY 
dx ~ dy ~ dz ’ dz + dy ~ ’ dx dz ~ ’ dy dx ~ ^ ' * (**) 
0Y_0Z 9 _X_0X 
dx ~ ds ’ dz ds ’ dy ds, * * * * (®) 
The equations (4) are of course satisfied by E = grad V, where Y is a 
solution of Laplace’s equation 
02y 0 2 y 02y 
0X 2 + 0Y 2 + 0Z 2==O - 
Each component of E will then be a solution of Laplace’s equation, and 
so if equations (5) and (6) are also satisfied we shall have a solution of (3) 
* Electrical and Optical Wave Motion , pp. 124-127. The theorem is given here in a 
slightly different form. 
