I always laugh about the T3 - designed to be longer with a greater capacity until someone realised it couldn't get off the ground! So they put in a fourth mini-engine with an on/off switch (!). It basiclly has two settings; full power for take-off and "off"! Talk about Heath Robinson!!
It was worse than that; quite a sad saga that explains quite a lot about why we Brits weren't very good at selling aeroplanes outside UK.
In the beginning (of this saga) Hawker Siddeley had on the drawing board the HS121. It was about T3 sized, and planned to use three RR Medway engines. It might have been a viable competitor to the later B727, taking the market that went to that aeroplane.
But BEA said "it's far too big. Make it titchy and we'll buy some". So they did, the ludicrously undersized T1. Later they upgraded that to the T2 at BEA's request, about the limit for Speys (the Medway was never produced once the HS 121 was ditched).
Eventually BEA woke up to the size of aeroplane they should have bought in the first in the first place, and asked HS to make the T2 bigger yet.
So they stretched the T2 and added the RB162 boost engine, a compact turbojet that had been developed as one of many lift engines for vertical lift in the days before the brilliant Harrier defined that technology.
So the T3 was an abomination. Four engines (so not a TRIdent anymore?), five if you include the APU.
Now if BEA had simply ordered the original HS121 in the first place.... Or if HS had stuck to their guns and marketed their 121 internationally despite BEA.....