Readers of śāstric Sanskrit will know prasaṅga most of all as the "unwanted consequence". Something that follows automatically. Investigating in its past may shed some light also on this meaning.
prasaṅga is not extremely frequent in the Śrautasūtras and I failed to find a definition of it. Hence, one has to reconstruct its meaning indirectly, through its usage in the Śrautasūtras, through its usage in Grammar and Mīmāṃsā and possibly also through its etymological meaning. Every translation is therefore intrinsically tentative.
The following is my (provisional) hypothesis:
In the Śrautasūtras, prasaṅga might denote:
- 1. temporary and incidental association of two elements or rites
- 2. chance, i.e., occasion, for an extended application caused by this temporary association
- 3.what happens automatically, through a transport from one rite to another, unless there is an opposite prescription, in similar cases and if need arises
On tantra and prasaṅga, see here and here. On the methodological problem of reconstructing the meaning of a word, see here.