Who would have succeeded Chester A. Arthur if he had died just after becoming president?

[Joe Republic]

Chester A. Arthur became president on September 19, 1881 after the death of James Garfield.

The order of succession at the time only stipulated the Vice President, followed by the President Pro Tempore of the Senate, and then the Speaker of the House.  That was it.

The 47th Congress had convened for a special session from March to May 1881, but did not elect any leaders.  It wasn't until they reconvened for another special session on October 10th that a new PPT was elected, and not until December 5th that a Speaker was elected.

So for the three weeks between 9/19 and 10/8, President Arthur had literally nobody next in line to replace him.  Given that he already suffered from kidney disease, and assassinations could happen at any time as his predecessor found out, this was a particularly risky situation.

What would have happened if Arthur had suddenly died during those three weeks?