Where exactly does Maxwell predict longitudinal and transverse EM waves? In what work, on what page does J. Maxwell predict EM waves, what are his exact words?
Here is a relevant quote:

James Clark-Maxwell predicted the possibility that electromagnetic
waves might exist. In theoretical discussions designed to more
thoroughly explain his mathematical descriptions, Maxwell asked his
readers to consider two different kinds of electrical disturbance
possibly existing in Nature. The first consideration dealt with
longitudinal electric waves, a phenomenon which required alternating
concentrations of electrostatic field lines. This densified and
rarefied pulsation of electrostatic fields necessarily demanded a
unidirectional field, one whose vector was fixed in a Singular
direction. The only variable permitted in generating longitudinal
waves was the concentration of he field. Subsequent propagation along
the electrostatic field lines would produce pulsating thrusts on
charges, pulsations moving in a Single direction. These "electrical
soundwaves" were rejected by Maxwell, who concluded that such a
condition was impossible to achieve. His second consideration dealt
with the existence of transverse electromagnetic waves. These required
the rapid alternation of electrical fields along a fixed axis. Space
spreading electrical lines would supposedly "bend to and fro" under
their own momentum, while radiating away at the speed of light from
the alternating source. Corresponding forces, exact duplicates of the
alternations produced at the source, would be detected at great
distances. He encouraged that experimenters seek this waveform,
suggesting possible means for achieving the objective.
("Secrets of Cold war technology" by G. Vassilatos, ch.1 ).

I'm especially interested in Maxwell's own words about the first consideration, the longitudinal electric waves, unidirectional pulsations or "electrical sound-waves".
 A: From A Dynamical Theory of the Electromagnetic Field by James Clerk Maxwell, published in 1865, original full text here (boldface is mine):

The equations of the electromagnetic field, deduced from purely
experimental evidence, show that transversal vibrations only can be
propagated. If we were to go beyond our experimental knowledge and to
assign a definite density to a substance which we should call the
electric fluid, and select either vitreous or resinous electricity as
the representative of that fluid, then we might have normal vibrations
propagated with a velocity depending on this density. We have,
however, no evidence as to the density of electricity, as we do not
even know whether to consider vitreous electricity as a substance or
as the absence of a substance. Hence electromagnetic science leads to
exactly the same conclusions as optical science with respect to the
direction of the disturbances which can be propagated through the
field; both affirm the propagation of transverse vibrations, and both
give the same velocity of propagation. On the other hand, both
sciences are at a loss when called on to affirm or deny the existence
of normal vibrations.

Personal interpretation (please correct me if I am wrong): "electromagnetic science" stands for what is currently known as "Maxwell equations", while "optical science" refers to the experimental investigations in optics available at that time.
More on this (including details on the experimental confirmation of Maxwell's ideas by Hertz and others) can be found in "Maxwell, Hertz, the Maxwellians, and the early history of electromagnetic waves" by Sengupta & Sarkar, 2003.
