These are like stopwatches - normally the hands are still; push the crown and the second hand starts; push it again and it stops; push yet again and the second hand returns to zero. Elgin made them in two different types - a 60 second timer, like the one on the left; and a 10 second timer like the one on the right. In the former, the second hand takes 60 seconds to go around the dial, whereas in the latter, it takes only 10 seconds. More on that later.
Elgins Timers differ from a standard stop watch in that while the balance and train on a stop watch run only when timing, the balance and train on the Elgin Timers runs as long as the piece is wound, with the hands only engaged and running when the crown is pushed.
The movements appear at first glance to be ordinary 16s Elgin pocket watch movements. The 60 second timer on the left is a 7-jewel movement, grade 469. This one dates to 1919. The 10 second timer on the right is a 15-jewel 582, from 1942. The gilt plates and plain finish mark this as a WWII piece - the War Department wouldn't pay for frills, apparently, and I also understand that nickel, normally used to coat the plates as in the 469, was needed for the war effort. You'll also see gilt plates in some of the wrist watches produced at that time.
It looks like the 10 second timer has no balance, but it's there! The 10 second timer runs at 30 beats per second, 6 times the speed of a normal watch. To accomplish this, Elgin combined a small, light balance wheel with a strong hairspring.
There it is, whizzing away underneath the regulator! These 10 second timers are often called 'Jitterbugs', because of the sound they make - a rattling buzz, kind of like an insect. Although some of the 15j 582s were 60 second timers, most seem to be the Jitterbugs. Once you see one running, you know why they didn't make the Jitterbugs in 7-jewel - they run so fast that they'd burn through their bushings in no time!
Okay, so how do they work?
What Elgin did was to add a column wheel chronograph-style stop works to the dial-side of a 16 size movement.
Compare this to the chronograph works on this Hamilton Model 23 chronograph.
Pushing the crown pushes down on the Actuating Lever at 1. It pivots at 2, so that the distal end at 3 is pushed out, pulling the Cam Hook at 4 to the left. The Cam Hook rotates the Column Wheel at 5 clockwise. The column wheel is topped with 5 columns and 5 spaces. Three levers are worked by these columns and spaces, as you'll see.
When the crown is pushed once,the column wheel rotates so that the end of the Connecting Lever at 1 falls into a space, allowing it to pivot up. This causes the Connecting Lever Pinion at 2 to move a tiny distance up. The Connecting Lever Pinion is meshed with the 4th wheel on the train side so it rotates whenever the train is running. The upper end moves into engagement with the Seconds Wheel, turning it at the same rate as the 4th wheel.
At the same time, the Flyback Lever at 3 engages a column, pushing it up so that the Minute and Second Flyback Hammers at 4 are lifted off the Heart Cams on the Second and Minute Wheels., allowing these wheels to turn.
The minute register does not turn smoothly, but rather jumps one space as the second hand passes Zero. The Second Wheel has a Finger Piece at 1 which turns the intermediate wheel at 2 as it passes, once per rotation. The Intermediate Wheel meshes with the Minute Register Wheel, so it turns it one click, limited by the Pawl Spring at 4.
A second push of the crown rotates the Column Wheel another step, causing a column to push the Connecting Lever down at 2, disengaging the Connecting Lever Pinion at 2 from the Second Wheel, stopping the hands.