"Since the shadows are similar in length to the girls heights, any object (such as a ball) roughly 3x their height in the air would cast a shadow on the ground that would be at least 3x it's physical size."

HUH?!?
I think what you meant to write was that the shadow would be distant approximately 3X the height of the girls (or equal to the height of the object), in the same direction as the girls' shadows. The size of the shadow is almost unchanged by its height, barring blurring of the edges due to the finite visible diameter of the Sun (penumbra shadow.)

That would put the "ball's" shadow roughly on a line from midway between the girls, to the nearby tree. And unless it's behind the girl on the right, I don't see it, either.

Dave B

I have made several observations, and come up with the following theory: There isn't any ball or anything else in the air; the girls were simply dancing around, as young girls of that era were prone to do on a warm sunny day. I come to this conclusion based on the length of the shadows left by the girls. Since the shadows are similar in length to the girls heights, any object (such as a ball) roughly 3x their height in the air would cast a shadow on the ground that would be at least 3x it's physical size. The only thing between them that resembles a shadow is a small dark patch (nearer to the girl on the left), that appears to be 4-5 inches in diameter. Given that, the ball they threw in the air would most likely have to be less than 2 inches in diameter to cast that specific shadow. Therefore; unless they were throwing marbles or superballs in the air, I postulate they weren't throwing anything at all. I base my theory on strict principles of quantum physics, certain geometric theorems, orbital mechanics, lightwave theory, and a big, fat, uneducated guess. Top that, Einstein!

[It has already been established that the exposure would not have been fast enough to stop a thrown object in motion, thus said object would register as a blur. It therefore follows that any shadow of the object would also register as a blur, and further, would be virtually undetectable when cast against such a background as the grass. Hence, the apparent absence of a shadow is inconclusive as to the existence or non-existence of the object. -A.E.]

Thank you Mr Einstein. ;-) Also, Dave B. At the time of the morning when I wrote this, thoughts were not guaranteed to be coherent. :-P

I think this might be it. A vertical, vaguely cylindrical blur.

I may have found it—the object of the girls' attention that is. I've circled a black, disk-shaped item in front of one of the bushes in the background. Dave may be able to clarify on his hi res version. This item is definitely not part of the shrubbery. Looks about 2 inches in diameter. I too am spending a little too much time on Shorpy, but I love it!

[That's a gap in the leaves. - Dave]

Thanks for the hi res clarification, Dave. Looks like we may never know!!

[Scroll up. - Dave]

I am sure they have spotted a balloon-borne Winnie-the-Pooh on his way to get the honey. Grownups can't see him.

They're trying to find the mysterious orb that's hovering to the right of the bench-sitters. Or maybe this is a photo version of the old "make the rubes look up" trick. It worked.

The girl on the left is the spitting image of Nancy from Little House on the Prairie. Nellie Oleson's adopted younger sister.

From their postures and the blurring of their arms it looks like the girls could be playing toss-and-catch with a single diabolo, or each tossing and catching their own diabolos, but in close-up their blurred hands don't appear to be holding the whip handles. If the handles were very thin, as was sometimes the case, they might have disappeared in this slow exposure, but it's hard to make a good case for it.

The girls appear to be twirling a skipping rope between them. You can faintly see a blur that appears it couold be the end of the rope in here hand.

Another theory is that they are not tossing something back and forth, but rather each individually up into the air.

But at first glance I thought they are playing diabolo.
(My grandmother, born in 1905, teached my kid how to play it - at 97!)

Keep in mind that Detroit Publishing was not in the candid-snapshot business; the end product here would be a color postcard, very possibly with the ball added in post-production.

is that my eyes aren't what they once were. As tterrace noted, the girls aren't facing quite the right way to be looking at the birdhouse and people. Still, before they wandered off toward the birdhouse and bench-sitters, those laser-tag lines were originally approximations of the girls' apparent lines of sight (see attached), and I couldn't find anything at all in the highlighted area to suggest an object at or near their depth (in particular, no blur), even playing with false color and contrast and such. Oh, well. Maybe some younger, sharper eyes can provide us with an answer. Or maybe these girls are the forebears of the little boy in the Sixth Sense movie. Or maybe, just maybe, I'm spending a little too much time on Shorpy these days.

I was able to find out what the girls were tossing back and forth.

The problem with the birdhouse theory is that the girl on the left would have to be looking toward her left for it to be in her line of sight; in fact, she's looking slightly toward her right relative to the tree. As for the girl on the right, she's looking up rather than across toward the people on the bench across the way. I think it's clear they're both looking at whatever object they're tossing back and forth. And since the exposure was not fast enough to freeze their arms in motion, an object in flight would also be a blur; less so if it happened to be caught at the top of its arc of trajectory, but still a blur rather than a crisp image.

It may be that, but I agree that the two gals are playing some kind of pitch and catch with a ball that is invisible to the slow speed of the camera lens.

First glance at Splunge's version and I thought they were playing laser tag. Pinafored warrior on the left is a horrible shot, says I to myself. "You don't have to use any windage with lasers," I wanted to tell her. Then I realized it was too late; by now she'd be dead anyway.

There's a birdhouse (or something like one)at about 4 branches up on the very right.

These young ladies are clearly pointing to one of the many UFOs that plagued Columbus during the summer of 1908.

Just visible in the trees.

[I don't think so. - Dave]

If one selects a point pretty much equidistant between the girls feet, and then lifts one's gaze vertically, there is what appears to be a ball hovering.

I suppose those girls would now be around one hundred and twelve years old. I hope they are still having fun playing ball somewhere.

First impression of the person on the park bench at right was that it was a man with an upside-down umbrella. However, upon closer inspection, it appears to be a woman with a HUGE hat. Sally Fields would indeed be jealous.

There is something there - or rather, two somethings. It looks to me like the girl on the left has noticed the birdhouse high in the tree to the right, and is gesturing toward it ("Look! A birdhouse!"), while the girl on the right is waving at the people on the shaded bench to the left. Or maybe there's really more here than meets the eye.