פירוש על ראש השנה 4:12

The [variant text] "What does this teach us?" is in error rather the text is "this teaches us that Nissan is the New Year for Kings..."

Where they appointed him - the princes met and concluded that he should be appointed

For both of them - Someone who writes a document in the Adar after the first's ascension can write either in year x of the dead king or can write in the first year of the new king

He died in Nissan and a successor took over in Nissan [of that same year] - and it is the law that if he ascended in any month that until the coming Nissan can use both the count of the dead king and that of the new one.

and the second year to the second king - He who writes "from x's ascension" does not count the second year to the dead king because this year was not his and the first isn't counted to the second [king] because he hadn't ascended then rather the second is counted as the second [king's] first year.

You might think you couldn't count 2 kings for one year - [your initial thought might be] that since the king didn't reign for long the entire year should be accredited to the first dead king.