The text:

Original Kanbun:

諸大名参勤作法之事
続日本紀制日　預不公事恣不得集己族　京裡二十騎以上不得集行云々　然則不可引卒多勢　百萬石以下二拾萬石以上不可過二十騎　十萬石以下可為其相應　蓋公役之時者可随其分限矣

Yomikudashi:

諸大名参勤作法ノ事。
続日本紀制シテ日ク、公事ニ預ラス、恣ニ己ガ族ヲ集ムルヲ得ズ。京裡二十騎以上集行スルヲ得ズ云々。然レハ則チ多勢ヲ引卒スベカラズ。百万石以下二拾万石以上二十騎ヲ過クベカラズ。十万石以下ハ其レ相応タルベシ、蓋シ公役ノ時ハ其分限ニ随フベシ。

Translation:

The rule all daimyōs must follow when reporting for duty:
The rule as recorded in Shoku Nihongi says that unless entrusted with official duty one is not allowed to gather his clansmen as he wishes, and no more than twenty horsemen are allowed to gather in the capital. Therefore one is not permitted to lead a large force. Those below 1,000,000 koku and above 200,000 koku are not not allowed more than twenty horsemen. Those 100,000 koku and below are to have a proportional number. However, when on official business, one is to [have a number] appropriate to his rank.

The English translations I have on hand (including Wikipedia) interprets "capital" as Edo. However, as the article references Shoku Nihongi, and Edo was not the term for "capital" at the time (if ever), the current academic consensus in Japan is that the "capital" is in fact Kyōtō. There's some disagreement on whether this would be to the Imperial Palace (the Emperor) or Fushimi Castle (the Edo Bakufu), but either way the regulation is for when reporting to Kyōtō.

As you can see above, the attendance part of alternate attendance goes way back in Japanese history to at least the Heian, if not the Nara period. Both Kamakura and Muromachi Bakufu had their own attendance rules, called the banyaku in the Heian and Kamakura. However, this article does not lay out the rule for the alternate (kōtai) part of alternate attendance for Edo Bakufu. That does not appear in Bakufu law until article 2 of the 1635 version:

Original Kanbun:

大名小名在江戸交替所相定也　毎歳夏四月中可致参勤　従者之員数近来甚多　且国郡之費且人民之労也　向後以其相應可減少　但上洛之節者任教令　公役者可随分限事　

Yomikudashi:

大名・小名在江戸交替相定ムル所ナリ。毎歳夏四月中参覲致スベシ。従者ノ員数近来甚ダ多シ、且ハ国郡ノ費、且ハ人民ノ労ナリ。向後ソノ相応ヲ以テコレヲ減少スベシ。但シ上洛ノ節ハ教令ニ任セ、公役ハ分限ニ随フベキ事。

Translation:

It is ruled that daimyōs and shōmyōs [small daimyō] are to alternate their location with Edo. They are to report for duty in the summer during the fourth month of every year. Lately the number of followers [to Edo] have been excessive. This depletes the provinces and districts, and exhaust the people. From hence forth, the number is to be appropriately reduced. However, when reporting to the capital, one is to follow the rules [assumed to be article 9 of the 1615 version above] and when on official business to [have a number] appropriate to his rank.

For the purpose of this question, note that the article calls Edo "Edo", and jōryaku is used for going to the capital (Kyōtō), with the same rules as given in article 9 of the 1615 version.

From this we can see that by 1635, some sort of alternate attendance was de facto in place. It seems that prior to this, various daimyōs simply did it on their own accord to demonstrate their loyalty to the Edo Bakufu, as it was already the tradition for samurai to spend considerable amount of time in the service and have their family live at their lord's castle town, and, as already mentioned, attendance laws go way back in history. The 1635 law simply sets out how it was supposed to be conducted henceforth.