तद्गच्छति पथिदूतयोः

Operational rule vidhi

An affix -- प्राग्दीव्यतोऽण् (4.1.83) etc. comes after a word in the Accusative -2nd case in construction in the sense of 'what goes thereto', provided that, which goes, is a road or a messenger.

Word by word (padaccheda)

Substitute / term तत् prathamā · ekavacanaVerb गच्छति tiṅantaRight context पथिदूतयोः saptamī · dvivacana

Substitute / term Prathamā (1st case): in operational rules, the substitute or the thing introduced; in definitions, the term or its referent.
Verb A finite verb form (tiṅanta).
Right context Saptamī (7th case): "when X follows", the operation applies to what precedes (1.1.66 तस्मिन्निति निर्दिष्टे पूर्वस्य). Also a condition or domain ("in the sense of").

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Full reading

तत् गच्छति पथिदूतयोः

Pāṇini leaves out words that carry over from earlier sūtras. Expand to see the rule as a reader actually understands it.

Commentary

An affix -- प्राग्दीव्यतोऽण् 4.1.83 etc. comes after a word in the Accusative -2nd case in construction in the sense of 'what goes thereto', provided that, which goes, is a road or a messenger.
Thus स्रुघ्नम्गच्छति = स्रौघ्नः "a road or a messenger that goes to Srughna." So also माथुरः ॥ The road may either lead to Srughna, or being in Srughna, be used for going.

Why do we say पथिदूतयोः "meaning a road or a messenger". Observe स्रुघ्नं गच्छति सार्थः 'he goes to Srughna for his own sake'.

Śrīśa Chandra Vasu, The Ashtadhyayi of Panini (1897), via ashtadhyayi.com.