Page:BatemanConformal.djvu/6

1908.] Accordingly, from a solution V = F(x, y, z, w) of the equation

we may derive a second solution

and from the solution U = f(x, y, z, w) of

we may derive another solution

Putting w = ict, where c is the velocity of light and t is the time, the equations take the well known form It may be mentioned here that Lorentz's fundamental equations of the electron theory, viz.,

may be reduced to a symmetrical form by writing s = ict and putting

The four mutually orthogonal vectors (A, B, C, D) whose components are respectively

(0, r, -q, -p), (-r, 0, p, -q), (q, -p, 0, -r), (p, q, r, 0)

satisfy the equations

where

$div\ M\equiv\frac{\partial M_{1}}{\partial x}+\frac{\partial M_{2}}{\partial y}+\frac{\partial M_{3}}{\partial z}+\frac{\partial M_{4}}{\partial s}$ if $M\equiv\left(M_{1},\ M_{2},\ M_{3},\ M_{4}\right)$|undefined

Again, if we put

and introduce four new vectors $$A_{1},\ B_{1},\ C_{1},\ D_{1}$$ whose components are respectively

(S, -Z, Y, -X) (Z, S, -X, -Y) (-Y, X, S, -Z) (X, Y, Z, S),

we find

where

Finally, if X, Y, Z, S can be derived from a potential function n so that

we can form four mutually orthogonal vectors θ, Φ, ψ, χ whose components are respectively

(n, r, -q, -p). (-r, n, p, -q), (q, -p, n, -r), (p, q, r, n),

and the equations then take the simple form

and the transformation may be written

where now

The study of this transformation will be taken up later.