His incompetence and lowly status contributed to two of the most infamous convict ship voyages in the history of Australia.
Voyage of the Scarborough in 1790
He was probably employed by the contractor rather than the Navy on the voyage of the Scarborough in 1790. Under command of John Marshall, the Scarborough departed Portsmouth on 19 January 1790.
About 12th February plans of mutiny were whispered however they came to nothing when one of the prisoners, Samuel Burt informed the Captain of the plot. Prisoners were severely punished by flogging and were confined and chained to the deck. By the time they reached the Cape many were already seriously ill.
Perhaps Jacob Beyer had little influence over the regulations and decisions on the voyage. He must have have been overwhelmed by the number of dead and dying prisoners.
He informed Naval Agent Lieut. Shapcote of the Neptune of the dire situation of the convicts when they reached False Bay in April:
Surgeon Beyer to Lieutenant Shapcote
15 April 1790
I am sorry to inform you that the scurvy is making a rapid progress, both amongst the soldiers and convicts; of the former I have ten, five of which are very bad, and of the latter there are fifty-seven, whereof twenty three are confined to their beds. It is my opinion that fresh provisions and greens every day may be procured while we are in the harbour; by this means will be saved a vast number of them from falling a sacrifice to that I have et. August Jacob Beyer, Surgeon 
In The Convict Ships, Charles Bateson wrote: - - The rations of the prisoners in the Scarborough were not deliberately withheld but owing to the reported mutiny, convicts were very closely confined. It was to this fact that the high death rate was directly due. 
By the time the Scarborough reached Port Jackson on 28 June 1790 seventy-three prisoners had died.
Judge Advocate David Collins described the scene on the landing of the prisoners: - On the evening of the 28th the Neptune and Scarborough transports anchored off Garden Island, and were warped into the Cove on the following morning. By noon the following day, two hundred sick had been landed from the different transports. The West side afforded a scene truly distressing and miserable; upwards of thirty tents were pitched in front of the hospital; all of which as well as the adjacent huts, were filled with people, many of whom were labouring under the complicated diseases of scurvy and dysentery, and others in the last stage of either of those terrible disorders, or yielding to the attacks of an infectious fever. The appearance of those who did not require medical assistance was lean and emaciated. Several of these miserable people died in the boats as they were rowing on shore, or on the wharf as they were lifted out of the boats; both the living and the dead exhibited more horrid spectacles than had ever been witnessed in that country. 
A female convict who came out on the Lady Juliana described the scene when the Second Fleet arrived in Port Jackson.....
Oh! if you had but seen the shocking sight of the poor creatures that came out in the three ships it would make your heart bleed; they were almost dead, very few could stand, and they were obliged to sling them as you would goods, and hoist them out of the ships, they were so feeble; and they died ten or twelve of a day when they first landed; but some of them are getting better. The Governor was very angry and scolded the captains a great deal, and, I heard, intend to write to London about it, for I heard him say it was murdering them. 
Voyage on the Boddingtons
Perhaps Beyer wasn't held accountable for the deaths on the Scarborough for in 1792 he was once more engaged as Surgeon. This second voyage was very different in terms of the health of the convicts. The Boddingtons departed Cork on 15 February 1793 and arrived in Port Jackson on 7 August 1793. One hundred and twenty-five male prisoners and 20 female prisoners were embarked and only one prisoner died on the voyage.
Charles Bateson in The Convict Ships (pp.44-45) included part of Naval Agent Richard Kent's report to the Home Department in 1793:
I must say that it would be right to bind down the captains of ships carrying convicts under the direction of an agent, that he might comply with the orders given him for the preservation of the lives and health of the convicts; for, if I had not persevered and got everything done myself on the Boddingtons, for the cleanliness and comfort of the convicts, I do believe there might be a great mortality amongst them; for my orders respecting them were never attended to, and Captain Chalmers told me he only came in the ship to navigate her. After which I contrived to get the convicts themselves to preserve order, cleanliness and regularity among one another, and I am happy to say that the trouble I took in keeping them in order was amply compensated in the little trouble there was with them in the medical department.
Voyage of the Britannia
Augustus Beyer's third convict ship voyage was on the Britannia in 1797.
His appointment to the Britannia was at the last minute. Either Beyer was not held in any way responsible for the disaster of the Scarborough seven years previously or it was overlooked as the following correspondence shows:
Lord Castlereagh to Under Secretary King....
Dublin Castle 18th September 1797 (Extract),
I beg leave to observe that the person appointed here to go out as surgeon to the convicts in the Britannia declined to proceed on the voyage just as the vessel was ready to sail, and the business was undertaken by a Mr. Beyer, who came from England to Cork in that ship, and who, it was represented, had gone two voyages to Port Jackson with convicts with great success..
The voyage of the Britannia was one of horror for the convicts. She departed Cork on 10 December 1796 and arrived in Port Jackson on 27 May 1797. 144 male prisoners and 44 female prisoners were embarked and ten male and 1 female prisoners died on the passage out.
The harrowing story of the voyage of the Britannia under Captain Thomas Dennott is told in Charles Bateson's The Convict Ships..... As in the Second Fleet transport Neptune, the combination of a callous and brutal master and a weak, incompetent surgeon made the voyage of the first Britannia one of the worst in the history of transportation. There was one death to every 17 prisoners embarked, 10 men and one woman dying out of 144 men and 44 women; but the convicts were brutally mistreated and the survivors were landed in a wretched and emaciated state. The Britannia's master, Thomas Dennott, was a sadist who, in consequence, as Governor Hunter declared,' of some conjecture of mutiny', kept the prisoners confined in irons and flogged them unmercifully. Even the women received three or four dozen cuts from a cane for the most trivial offences.
Notes and Links
1) Beyer/ Beier?
 Bateson, Charles Library of Australian History (1983). The convict ships, 1787-1868 (Australian ed). Library of Australian History, Sydney
 Collins, David, An Account of the English Colony in New South Wales, p. 102
 HR NSW, vol 2, . Extract from a letter by one of the female convicts transported in the Lady Juliana. Sydney Cove 24 July 1790, p.767