For those who might be interested (apologies to those who aren't) here is jones's own account of the battle

"Information from prisoners had confirmed that the fleet was escorted by the Serapis, a new vessel that could mount 56 guns but then mounted only 44 in two batteries, one composed of 18-pounders, and by the Countess of Scarborough, a new frigate mounting 22 guns. When the enemy saw that we had taken the chase, the Serapis and the Countess of Scarborough took advantage of the wind to stand out to sea while the convoy crowded on sail toward the fortress of Scarborough.
"As there was little wind, I was unable to close with the enemy before night. The moon did not rise until 8 o'clock, and as soon as it was dark the Serapis and the Countess of Scarborough came about and put on all sail for the fort of Scarborough.
"I was fortunate enough to discover this enemy movement with my night glass, otherwise they would have escaped me. As this forced me to alter my course by six points of the compass with the intention of cutting off the enemy from their retreat toward shore, the captain of la Pallas concluded that the crew of the Bonhomme Richard had revolted, and this idea convinced him to haul his wind and to stand away from the shore. At the same time, the Alliance lay windward of the enemy at a considerable distance. Because the captain of this vessel had not paid attention to the signals of the Bonhomme Richard since leaving France, I was obliged to run all the risks and engage the enemy with the Bonhomme Richard alone to prevent their escape.
"I began the battle at 7 o'clock at night and within pistol range of the Serapis, and I sustained it for nearly an hour at that distance, exposed at the same time to the attack of the Countess of Scarborough, which raked the stern of the Bonhomme Richard with broadsides.
"It has been noted that properly speaking the Bonhomme Richard was only a frigate of 34 guns, the battery of which was of 12-pounders but that it had been decided to mount six 18-pounders above the powder magazine, which would have been very useful in cannonading a port. The sea was very calm during the battle with the Serapis and I hoped to derive a great advantage from these six 18-pound guns. But instead of that, the old cannons burst at the beginning of the action and the officers and men above the powder magazine, who had been selected as the best of the crew, were killed, wounded, or so frightened that none of them was of any use during the remainder of the engagement.
"In this unfortunate extremity, having to contend with forces three times superior to my own, the Bonhomme Richard was in great danger of going to the bottom. With her battery out of action I had recourse to the dangerous expedient of throwing grappling irons on the Serapis in order to nullify the superior power of her two batteries and to shield myself from the fire of the Countess of Scarborough. This maneuver succeeded perfectly, and with my own hands I tied the Serapis to the Bonhomme Richard. The captain, Thomas Piercy, of the Countess of Scarborough, an illegitimate son of the Duke of Northumberland, conducted himself like a man of sense and from that time on ceased fire on the Bonhomme Richard, knowing full well that he could not damage us without equally damaging the Serapis.
"The Serapis being then to windward, dropped her anchor as soon as she was hooked, hoping by this to disengage herself from the Bonhomme Richard, but success did not answer her expectations. From then on the combat was limited to the firing of cannon, swivel guns, muskets, and grenades. The enemy at first showed a desire to board the Bonhomme Richard; however, after having thought it over, they did not dare try. But the Serapis had the advantage of her two batteries, besides the cannon on the quarterdeck and on the forecastle, whereas the Bonhomme Richard's cannon were either broken or abandoned, except for four pieces on the quarterdeck, which were also abandoned for some minutes. The officer, Matthew Mease, who commanded these four cannons on the quarterdeck was dangerously wounded in the head, and having at that moment no object more deserving of my attention, I took command of them myself. some sailors came to aid me of their own accord and serviced the two cannon alongside of the enemy with surprising skill and courage. A few minutes later I found enough men to transport one of the cannon on the quarterdeck to the opposite side, but I was not able to find sufficient force to bring the other, so I could bring to bear only three guns against the enemy for the rest of the action.
"The moon rose at 8 o'clock in the evening and the two vessels were then in flames from the cannonade. That was why the Serapis' mainmast, which was painted yellow, was such an easy object to distinguish, and I pointed one of my guns loaded with bar shot at it. In the meantime the two other pieces were well used to destroy the barricades of the enemy and to sweep their quarterdeck with oblique fire. Only the men on the topmast bravely supported the quarterdeck cannons with muskets and swivel guns and threw grenades on board the enemy vessel with great skill. In this way the enemy were killed, wounded, or driven from their stations on deck and aloft, notwithstanding the superiority of their artillery and manpower.
"Captain Richard Pearson of the Serapis consulted with his officers and they resolved to surrender, but an unfortunate circumstance happened on board the Bonhomme Richard to prevent them. A bullet having cut one of our pumps, the master carpenter, John Gunnison, was seized with panic and cried to the chief gunner, Henry Gardner, and the master at arms John Burbank, that the Bonhomme Richard was sinking. This idea so terrified these men that they forgot their duties and thought only of saving their lives. At the same moment, someone told the chief gunner that the lieutenant and I had been killed. As a result, thinking that he had become the commanding officer, the chief gunner rushed to the bridge to haul down the American flag, which he would have done if the flagstaff had not been carried away when the Bonhomme Richard hooked the Serapis.
"The captain of the Serapis, hearing the chief gunner of the Bonhomme Richard ask for quarter because he thought the Bonhomme Richard was sinking, hastened to cry to me: "Do you ask for quarter? Do you ask for quarter?"
"I had been so occupied in firing the three cannon on the quarterdeck, I did not know what had passed between the chief gunner, the master carpenter, and the master at arms, so that I replied to the English captain: "Je ne songe point a me rendre, mais je suis determine a vous faire demander quartier."*

*(webmasters note - Allow me to interrupt here, kind reader, to point out that these are Jones' EXACT words from the original manuscript in the Library of Congress. Translated, it becomes: "That point didn't occur to me, but I am determined to make you ask for quarter." "The 1812 translation that most of this webpage comes from translated it as: "I do not dream of surrendering, but I am determined to make you strike." That last phrase is a reference to striking colors, or the lowering of a ships' national flag as a sign of surrender. It wasn't until 1825, some thirty three years after Jones' death, that a biographer very loosely translated it as: "I have not yet begun to fight.")

"The captain of the Serapis, however, conceived some hope, because of what the American chief gunner had said, that the Bonhomme Richard was about to sink. But when he found that his men on the upper decks were in imminent danger, he sent them to the main deck to service the two batteries, which they fired against the side of the Bonhomme Richard with the fury of vengeance and despair. It has been observed that, when I began the action, la Pallas was a great distance to the windward and the Alliance also lay to the windward. When the captain of la Pallas heard action begin, he approached and spoke to the Alliance, but they lost much time and it was not until after all that has been related that the two frigates came within cannon range of the Countess of Scarborough.
"Because Ia Pallas engaged this frigate while sailing before the wind and tide (at the same time that the Serapis was at anchor and under the grappling irons of the Bonhomme Richard, which had the wind astern), soon they were both a considerab]e distance to leeward. The Alliance followed Ja Pallas and the Countess of Scarborough and while passing along the exterior side of the Bonhomme Richard delivered a broadside within gunshot range against the bow of this frigate and the stern of the Serapis, which together formed one small target. But it is easy to suppose that the broadside of the Alliance did more damage to the Bonhomme Richard than to the Serapis because the men of the Serapis had been chased from the upper decks to the covered deck; whereas on board the Bonhomme Richard not only a number of people who were then on the upper decks (after they had been chased by the two enemy batteries from the places where they were hidden) but also the men who were serving the pumps and the three guns on the quarterdeck were much more exposed.
"The battle between the Bonhomme Richard and the Serapis continued with the greatest intensity. The bulwarks of the Serapis were damaged or burned, and the mainmast was gradually cut down by the grapeshot of the Bonhomme Richard, while the much superior artillery of the Serapis' two batteries struck one side of the Bonhomme Richard and blew out the other so that during the last hour of combat the shot passed through both sides of the Bonhomme Richard meeting little or no resistance. The rudder was shattered and only an old timber here and there kept the poop from crashing down on the gundeck.
"After a retreating action of short duration the Countess of Scarborough surrendered to la Pallas. They were then a considerable distance to the leeward of the Bonhomme Richard and the Serapis. The Alliance, which had followed them downwind, lost much time in going this way and that and questioning the prize and Ia Pallas, but finally the captain of la Pallas asked the captain of the Alliance: "Do you want to take charge of the prize, or go to aid the commodore?" and the Alliance began to maneuver to gain the wind. She tacked several times before regaining the wind and finally she sent a second broadside against the bow of the Serapis and the stern of the Bonhomme Richard. Some other people and I shouted to the Alliance to cease firing for God's sake and to send some men on board the Bonhomme Richard. The captain of the Alliance disobeyed; passing alongside the Bonhomme Richard and bringing a few cannon to bear during the passage, she unleashed a third broadside against the bow of the Bonhomme Richard and the stern of the Serapis. After this the Alliance kept at a respectful distance and took great care not to expose herself either to receive a blow or to have a single man killed or wounded.
"The idea that the Bonhomme Richard was going to sink had so deranged the master at arms mind by excessive fear that he opened the hatches and, despite my repeated orders to the contrary, let out all of the prisoners we had, numbering 100. At the time of outfitting, the commissioner had refused to provide iron chains for the prisoners, and this mental derangement of the master at arms might have become fatal, if I had not taken advantage of the prisoners' fear and put them to work at the pumps where they displayed surprising zeal, appearing to have forgotten that they were prisoners and that nothing could prevent their leaving the Bonhomme Richard to board the Serapis, as it was entirely in their power to put an end to the fight by killing me or throwing me overboard.
"As the three guns of the Bonhomme Richard continued to fire without interruption against the Serapis and finally cut down the railing on her quarterdeck and her mainmast, so that the latter was only supported by the yards of the Bonhomme Richard, and at the same time the men in the rigging maintained a continuous fire of muskets, swivel guns, and grenades, the enemy began to slacken their fire and soon lost all hope. One circumstance that contributed a great deal to the victory of the Bonhomme Richard was the extraordinary presence of mind and intrepidity of a Scottish sailor, William Hamilton, who was posted in the mainmast. This brave man, on his own accord, seized a lighted match and a basket of grenades and advanced along the main yard of the Bonhomme Richard until he was directly over the enemy's upper deck, and as the flames from their railings and shrouds added to the light of the moon he could see all that happened on the enemy vessel. Every time he saw two or three men gathered together he would throw a grenade among them. He was even skillful enough to throw several into their hatchways, and one of them set fire to the charge of an 18-pounder on the first gundeck, burning a number of people.
"At this point the captain of the Serapis advanced on the upper deck, lowered his flag, and asked for quarter. At the very instant that he was lowering his flag, his mainmast fell into the sea. He came with his officers from the Serapis onto the Bonhomme Richard and presented me with his sword. While this was happening 8 or 10 men of the Bonhomme Richard made off with the Serapis' shallop, which had been in tow during the fight.
"It was after 11 o'clock when the battle ended; consequently, it had lasted more than four hours. The Bonhomme Richard had on board only 322 men, good or bad, when the battle began; and the 60 men who were stationed in the powder magazine when the cannon burst, having been of no service during the action, cannot properly be counted as part of the force that opposed the Serapis. While in Denmark the Serapis had received a number of English sailors who had come from India to that country, so that according to the roll, which was found after the battle, there were more than 400 men on board when she first encountered the Bonhomme Richard.
"Her superiority in cannon was even greater, not to mention the intrinsic value of her artillery, which so completely surpassed that of the Bonhomme Richard that it would be very difficult to compare them.
"Thus, putting aside the damage done to the Bonhomme Richard by the Countess of Scarborough during the first hour of combat and by the three broadsides of the Alliance thereafter, it is not difficult to form a judgment on the combat between the Bonhomme Richard and the Serapis and on a victory obtained over so superior a force after such a long, bloody, and close range battle.
"La Vengeance, a corvette mounting 12 three-pounders, and the pilot's boat with the second lieutenant, Henry Lunt, of the Bonhomme Richard, another officer, and 18 men, could have been of singular service either in pursuing and capturing the convoy, or in reinforcing me by supplementing the men on board the Bonhomme Richard. But, strange to say, they remained all the time as spectators without interest in the affair, staying out of danger and to windward, and the least that one can say about the conduct of the Alliance is that it appeared to stem from a principle worse than ignorance or insubordination.
"It is clear from what has been said that if the Baltic fleet escaped, it is due particularly to the disorder that the commissioner created in the squadron through his avaricious cabals. And one can attribute the impossibility of waiting for the eight vessels from India and the fact that no enemy ports were destroyed or ransomed to this same cause.
"It is fair to say, however, that some of the officers who were on board the Bonhomnie Richard conducted themselves in a very admirable manner during the action. The lieutenant, Richard Dale, having been abandoned at the battery and finding that he could not rally his men, came up on deck, and, although wounded, supervised the working of the pumps. But despite all his efforts, the hold of the Bonhomme Richard was more than half filled with water when the enemy surrendered.
"During the last three hours of the battle, the two vessels were on fire. Quantities of water were thrown on it and the fire at times appeared to be extinguished, but it always broke out anew. After the action, it was thought to be entirely extinguished. The weather was calm during the remainder of the night, but when the wind rose a little the fire broke out again, much more dangerous for having penetrated the timbers of the Bonhomme Richard to within a few inches of the powder magazines. The powder was immediately carried on deck, ready to be thrown into the sea as a last resort. Finally the fire was completely extinguished by our cutting away planks and drowning it with great amounts of water.
"The next morning the weather was cloudy and foggy, and when it cleared around 11 o'clock all of the enemy convoy had taken refuge under the fortress of Scarborough and not a single sail was to be seen along the coast
. "We then examined the Bonhomme Richard to determine if it were possible for her to be conducted into some port. The examination ended at 6 o'clock in the evening, and we judged the thing impracticable, mostly because of blows she had received in the bow from the Alliance, causing holes that could not be closed. Consequent]y, I employed all the boats without delay to save the wounded by carrying them to other vessels This work took all night, and the next morning, despite our having continuously and vigorously employed the pumps, the water had entirely filled the hold. Then, as the wind rose, the Bonhomme Richard immediately sank. I saved only my signal flags. I lost all of my belongings, amounting to more than 50,000 livres, not counting a number of invaluable papers. The officers and men of the ship also lost all of their personal effects.
"I took command of the Serapis, on which we had jury rigged masts, but I was tossed about in the North Sea by contrary winds for 10 days before reaching the Texel. I would have liked first to debark my 600 prisoners at Dunkirk, and the wind was favorable for this enterprise the day I entered the Texel. But the commissioner's cabal opposed this necessary plan. Because that imprudent man had told the captains not only that the squadron was destined for the Texel, but also what its object was to be in Holland, they left me, and I was obliged to follow them into the Texel since they had most of the prisoners."