>>40550330
>>40550660
>>40550690
I should say, I would have preferred if you had posted a little bit more context, since this seems relatively clear now, but it didn't look that way when I first looked at it because the quotation starts off mid-sentence.
The rules followed here are basic planetary dignities, rulerships and dynamism. The planets rule various sections of the Zodiac, and here the dignities being used seem to be decans, signs (in the case of Mercury), and mansions (in the case of the Moon). These make the planet more powerful and able to create good and great things. Placing the planet on the Midheaven also makes it very active and dynamic, and means it actually makes the great things it is able to do. The petition rituals (I assume those are petition rituals given the lack of context) are set during those electoins, so long as it is also the planetary day and planetary hour of the relevant planet (four options are given).
Once again, I am not sure what the Southern Quarter stuff means. But the rest seems relatively intuitive and obeys basic rules of astrology (dignity and angularity) and magic (planetary days and hours). My initial thought was that the decan conditions were all required simultaneously which would be an incredibly impractical and obscure way of doing things, and would be impossible for me to break down the logic of such a requirement.