His analysis basically confirmed Herschel's picture: the MW is a
flattened elliptical system, with stellar density 
decreasing away
from the centre. drops to half its central value at 150 perpendicular to the MW plane, and 800 in the galactic plane. The
Sun is at 650 from the centre. So it appears as if the Sun is near
the centre of the MW. Since we've made this type of error once before
(assuming the earth was at the centre of the Universe), this made
several people slightly uncomfortable. And as we'll discuss later, it
turned out to be wrong!
In our derivation of counts of stars as function of distance 
, we
assumed that the luminosity drops as 
. One reason this could be
wrong, is because of absorption or scattering of the stellar light. For
Rayleigh scattering off atoms, there is a colour dependence. Of course,
not taking absorption into account would give you the wrong answer, and
Kapteyn was fully aware of this. If absorption were important, one
would expect fainter (more distant stars) to be on average
redder. Kapteyn found this indeed to be the case, yet the amount of
reddening was very small, and so absorption was no thought to be an
issue. The mistake was that the absorption is due to dust, not
Rayleigh scattering. Dust absorption is much less colour dependent,
and so the small reddening measured by Kapteyn actually implied a much
greater dimming of the distant stars.
The American astronomer Harlow Shapley, estimated the distances to the brightest globular clusters using their RR Lyrae variables, and found they were not centred around the Sun, but around a point 15kpc away in the direction of Sagittarius. This was in conflict with the Kapteyn universe - it seemed difficult to reconcile the Sun to be near the centre of the MW, yet the centre of the globular cluster system not to coincide with the centre of MW? There was no clear way how to reconcile these different data, and so the issue remained unresolved.
Meanwhile, another famous Dutch astronomer, van Maanen, found that some of the nebulae had proper motions, implying they were nearby, within the MW. (He compared the positions of the nebulae on plates taken several years apart). It was generally accepted these nebulae were probably proto-planetary systems, or something like that. van Maanen himself, but also Edwin Hubble later redid the proper motion measurements with better data (using a better telescope), and both failed to reproduce the proper motions. van Maanen probably just made a mistake! In the 1930s, Hubble made his even more famous discovery that the spectral lines of atoms in other galaxies are shifted to the red, by an amount proportional to the distance to the galaxy. The universe is expanding!
So it was a combination of mistakes (van Maanen), and not taking into account the importance of dust (Kapteyn), that led to the confusion. In the end, Hubble's better data clinched the matter. He resolved Cepheids in Andromeda, one of the bigger nebulae, and found a distance of 300kpc and so clearly outside of the MW. And so the size of the universe all of a sudden increased dramatically.
The 1920s were of course a period of great economic depression. But from the scientific point of view, they must have been tremendously exciting. In only a couple of years, the general theory of relativity was formulated, quantum mechanics revolutionised physics, and the Universe was found to be far bigger than realised before, and expanding as well.
It was not until the 1930s that Trumpler found what had been wrong with Kapteyn's results. He studied Open Clusters in the MW plane. He assumed them to be all of about the same size, and so got an independent relative distance for them. He used another estimate of their distance, based on `spectroscopic parallaxes' by observing the spectral type of some stars in the cluster, and using a calibration between spectral type and absolute magnitude. He showed that the more distant clusters (as judged from their angular size), appeared to have unusually faint stars in them. Or equivalently, that the more distant clusters (as judged from their photometric parallax) were larger. Clearly, something was wrong here. He correctly interpreted this, as due to the fact that the light from these more distant stars was strongly attenuated. By comparing the reddening with the attenuation, it was clear that a given amount of reddening corresponded to much more dimming than you'd expect for Rayleigh scattering. And hence Kapteyn's scales were wrong.
Before I summarise this in a time-line, just think of the fact that Einstein formulated his theory of general relativity, on which the current cosmological models are still based, before this puzzle was resolved. In fact, our view of the Universe around Einstein's time, was just plain wrong. I guess he rightly paid little attention to what the astronomers of his days told him!