Page:BatemanConformal.djvu/7

A second transformation of some interest is obtained by interchanging m with n and μ with ν. This changes

$-{\frac {1}{\nu }}f\left(-{\frac {l+\lambda }{2\nu }},\ -{\frac {l-\lambda }{2i\nu }},\ -{\frac {m+\mu }{2\nu }},\ -{\frac {m-\mu }{2i\nu }}\right)$

into

$-{\frac {1}{\nu }}f\left(-{\frac {l+\lambda }{2\mu }},\ -{\frac {l-\lambda }{2i\mu }},\ -{\frac {n+\nu }{2\mu }},\ -{\frac {n-\nu }{2i\mu }}\right)$

that is

f(x, y, z, w)

into

$-{\frac {1}{z-iw}}f\left(-{\frac {x}{2(z-iw)}},\ -{\frac {y}{2(z-iw)}},\ -{\frac {1-r^{2}}{2(z-iw)}},\ -{\frac {1+r^{2}}{2i(z-iw)}}\right).$

Putting w = ict and changing the sign of z, the formulæ for the transformation are

$X={\frac {x}{z-ct}},\ Y={\frac {y}{z-ct}},\ Z={\frac {r^{2}-1}{2(z-ct)}},\ cT={\frac {r^{2}+1}{2(z-ct)}},$

where, now,

$r^{2}=x^{2}+y^{2}+z^{2}-c^{2}t^{2}.$

If V = F(x, y, z, t) is a solution of

$\left({\frac {\partial V}{\partial x}}\right)^{2}+\left({\frac {\partial V}{\partial y}}\right)^{2}+\left({\frac {\partial V}{\partial z}}\right)^{2}={\frac {1}{c^{2}}}\left({\frac {\partial V}{\partial t}}\right)^{2},$

the function F(X, Y, Z, T) is also a solution, and, if U = f(x, y, z, t) is