At least to me, it was a bit mysterious just how the Hagans system of articulation worked. The Hagans system was designed in an effort to develop locomotives that had all the advantages of articulation in travelling on tightly curved track without requiring flexible steam joints, gear drives or more than the two cylinders of rigid framed locomotives. A number of different systems using rod connections were developed, the most successful being the Klose system and the Hagans system. Apparently these systems were only used on narrow gauge railroads. The Hagans system as used on the North East Dundas Tramway is actually fairly easy to figure out when you visualize the motion of the rods as the engine moves and the rear 0-4-0 assembly swivels.
The Hagans engine's three front drive axles are located rigidly in the frame and are powered off the cross head in the usual manner. The rear two drive axles are powered off a second "crosshead" that is moveable as the articulated section pivots in curves. (the NGDF did it again, turning my picture, but there is a right side up picture a few posts earlier)
This diagram shows the rod and lever system that transfers motion from the front crosshead to the back "crosshead'
The front drive lever is attached to the frame at one end and to the cross head at the other end. The front drive lever transfers motion to the rear drive lever via the drive lever coupling rod. The rear drive lever has the rear "crosshead" at the lower end and is connected to the counter steering lever at the other end. The counter steering lever is attached to a pivot on the frame and is connected to the counter steering rod on the lower end. The counter steering rod is connected to a bearing on the #4 axle. As the #4-#5 drive unit swings , it imparts
forward or rearward motion to the counter steering rod and to the bottom hinge of the counter steering lever. lever. Thus as the #4 axle moves forward, the counter steering rod moves forward, and moves the bottom end of the counter steering lever forward, with the result that the top of the counter steering lever pivots backward, which results in the gap between the drive lever coupling rod and the # 4 axle shrinking, thus compensating for the forward motion of the #4 axle. If you try to visualize the motions, it turns out to be pretty simple and pretty ingenious. As I understand it, the primary drawback of this scheme was the higher maintenance effort required to keep these engines going. So eventually the K1 and K2 Garratts won out.
Another view of the Hagans engine, possibly at the factory