Each team races the other twice in the head-to-head match race format, with a point for the winner of each race. The challenger with the fewest points at the end of the qualifiers will not go through to the semi-final round.
The British campaign has been far from smooth, with a collision with New Zealand during practice racing followed by a more damaging altercation with the Japanese team on Saturday which tore a huge hole in one of the British catamaran's hulls.
Although Ainslie's shore team managed to get the boat patched up and ready to sail again on the crystal-clear waters of Bermuda's Great Sound on Sunday, the British sailor and his team have now lost four of their five races.
The British crew's latest loss was significant as the French were widely regarded as the underdogs and had struggled with speed and stability on their first day, before securing victory on Sunday against Swedish team Artemis Racing.
FAST START
In Monday's other races, New Zealand beat Artemis in the most closely-contested duel of the event and Japan dominated France after a text-book start by skipper Dean Barker.
This means France, Japan and Sweden all have two points, leaving the event wide open.
"Its very much a learning process at this stage ... some of the good learning has come from the mistakes," Barker said after his crew's victory over Groupama.
The Artemis crew were fuming as they approached the finish line first but had to let New Zealand cross before them after the Swedish boat was penalised for not giving their opponents enough room to manoeuvre when they had right of way.
"You need room coming in at 40 knots," Emirates Team New Zealand helmsman Peter Burling said.
"We were fortunate to come away with it in the end."
In Land Rover BAR's battle with Groupama, Ainslie also dominated the start and looked to have built a commanding lead. But the French kept up the pressure and a poorly-executed manoeuvre by Ainslie's crew allowed Groupama skipper Franck Cammas to come from behind.
The sleek 50-foot (15 metre) foiling catamarans were neck and neck for much of the next leg, but Ainslie's team lost ground at another key moment as their boat came off its foils, allowing Cammas and his crew to gain a lead.
"Come on guys, let's go," Ainslie could be heard saying as he tried to get more from his crew and his high-tech boat.
"Keep working," Ainslie added.
But the French team kept their cool and delivered the speed and stability that eluded them in their first races, giving themselves a significant boost in their quest for the oldest trophy in international sport.
(Reporting by Alexander Smith in London; Editing by Ed Osmond)
