The affix yat comes in the remaining senses after the word arddha when preceded by 1. para 2. avara 3. adhama and 4. uttama,
Thus परार्द्ध्यम्, अवरार्द्ध्यम्, अधमार्द्ध्यम्, उत्तमार्द्ध्यम् ॥
Q. Why do we say पूर्वात् 'preceded by' ? Its use seems to be superfluous. If the sutra stood as परावराधमोत्तमेभ्यः, it would have given the same sense. For the अर्धात् being supplied from the last sutra, the whole sutra would have been परावराधमोत्तमेभ्योऽर्धाद् यत् which must mean 'preceded by para &c.'
A. The words पर and अवर have meanings other than that of दिक् or direction. As परं सुखं 'highest pleasure.' अवरं सुखं 'lowest pleasure.' Therefore, when these words पर and अवर denote direction, then by the subsequent rule, the affix would have been always ठञ् and यत् ॥ The employment of the term पूर्व in the sutra, prevents the application of the affix ठञ् even when the words denote direction.
According to the Vartika सपूर्वपदाट् ठञ् given under the last sutra, the word अर्ध preceded by any word would have taken ठञ्, hence the necessity of this sutra ordaining यत्, when the first terms are पर &c.,