Today we will look at the form आपगाः fNp from श्रीमद्भागवतम् Sb10.47.33.

एतदन्तः समाम्नायो योगः साङ्ख्यं मनीषिणाम् ।
त्यागस्तपो दमः सत्यं समुद्रान्ता इवापगाः ∥ १०-४७-३३ ∥

श्रीधर-स्वामि-टीका
तावता च कृतार्थो भवतीत्याह – एतदन्त इति । एष मनोनिरोधः अन्तः समाप्तिः फलं यस्य सः । समाम्नायो वेदः। स तत्र पर्यवस्यतीत्यर्थः । योगोऽष्टाङ्गः । सांख्यमात्मानात्मविवेकः। त्यागः संन्यासः। तपः स्वधर्मः। दम इन्द्रियदमनम्। मार्गभेदेऽप्येकत्र पर्यवसाने दृष्टान्तः – समुद्रान्ता आपगा नद्य इवेति ∥ ३३ ∥

Gita Press translation – “In the eyes of the wise, (the teachings of) the Vedas, the (eightfold) path of Yoga, the Sāṅkhya system of philosophy (which differentiates Matter from the Spirit), renunciation, suffering hardships for the sake of one’s faith, subjugation of one’s senses and truthfulness have their goal in mind-control even as rivers terminate in the ocean (33).

The प्रातिपदिकम् ‘आप’ used in the compound आपगाः is derived as follows –
अपां समूहः = आपम् – a collection of waters

(1) अप् आम् + अण् । By 4-2-37 तस्य समूहः – To denote a collection/group, the तद्धित: affix ‘अण्’ (prescribed by 4-1-83 प्राग्दीव्यतोऽण्) may be optionally applied following a syntactically related पदम् in which the सन्धिः operations have been performed, provided the पदम् ends in a genitive case affix and has a base denoting that of which the collection is intended.
Note: As per the सूत्रम् 4-2-47 अचित्तहस्तिधेनोष्ठक् – To denote a collection, the तद्धित: affix ‘ठक्’ may be optionally applied following a syntactically related पदम् in which the सन्धिः operations have been performed, provided the पदम् ends in a genitive case affix and has as its base a term denoting an inanimate object, or the term ‘हस्तिन्’ or ‘धेनु’।
Hence if waters were considered inanimate, as per the सूत्रम् 4-2-47, the affix ‘ठक्’ would have applied in the present example. By the fact that the affix ‘अण्’ (and not the affix ‘ठक्’) has been used to form the प्रातिपदिकम् ‘आप’, we can infer that waters are not considered inanimate.

(2) अप् आम् + अ । अनुबन्ध-लोपः by 1-3-3 हलन्त्यम्, 1-3-9 तस्य लोपः।

Note: ‘अप् आम् + अ’ gets the designation प्रातिपदिकम्‌ by 1-2-46 कृत्तद्धितसमासाश्च। This allows 2-4-71 सुपो धातुप्रातिपदिकयोः to apply in the next step.

(3) अप् + अ । By 2-4-71 सुपो धातुप्रातिपदिकयोः – A सुँप् affix takes a लुक् elision when it is a part of a धातुः or a प्रातिपदिकम्।

Note: The affix ‘अण्’ is a णित् (has the letter ‘ण्’ as इत्)। This allows 7-2-117 तद्धितेष्वचामादेः to apply in the next step.

(4) आप् + अ । By 7-2-117 तद्धितेष्वचामादेः – The first vowel among the vowels of a अङ्गम् which is followed by a तद्धित: affix (ref. 4-1-76 तद्धिताः) which is either a ञित् (has the letter ‘ञ्’ as a इत्) or a णित् (has the letter ‘ण्’ as a इत्) takes the वृद्धि: (ref. 1-1-1 वृद्धिरादैच्) substitute.

= आप ।

We can similarly derive the following –
१) काकानां समूहः = काकम् – a flock of crows
२) वकानां समूहः = वाकम् – a fight of cranes
३) वृकाणां समूहः = वार्कम् – a pack of wolves

The प्रातिपदिकम् ‘आपगा’ is a उपपद-समासः, derived as follows –
आपेन गच्छतीति = आपगा (नदी) – (a river) which goes (flows) with a collection of waters

(5) आप + टा + गम् + ड । By the वार्तिकम् (under 3-2-48 अन्तात्यन्ताध्वदूरपारसर्वानन्तेषु डः) अन्यत्रापि दृश्यत इति वक्तव्यम् – The affix ‘ड’ may be used after the verbal root √गम् (गमॢँ गतौ १. ११३७) when in composition with a उपपदम् which could be other than those specifically listed in 3-2-48 अन्तात्यन्ताध्वदूरपारसर्वानन्तेषु डः।
Note: In the above वार्तिकम्, the term अन्यत्र is in the seventh (locative) case. Hence ‘आप + टा’ gets the उपपद-सञ्ज्ञा here by 3-1-92 तत्रोपपदं सप्तमीस्थम्‌।

(6) आप + टा + गम् + अ । अनुबन्ध-लोप: by 1-3-7 चुटू and 1-3-9 तस्य लोपः।

(7) आप + टा + ग् + अ । By 6-4-143 टेः – When the अङ्गम् has the भ-सञ्ज्ञा, then its टि portion takes लोप: when followed by an affix that has the letter ‘ड्’ as an indicatory letter. Note: The अङ्गम् does not have the भ-सञ्ज्ञा here. But still टि-लोप: is done because otherwise no purpose would be served by having the letter ‘ड्’ as a इत् in the affix ‘ड’। डित्वसामर्थ्यादभस्यापि टेर्लोपः।

= आप + टा + ग ।

We form a compound between ‘आप + टा’ (which is the उपपदम्) and ‘ग’ by using the सूत्रम् 2-2-19 उपपदमतिङ् – A सुबन्तम् (term ending in a सुँप् affix) having the designation ‘उपपद’ (in this case ‘आप टा’) invariably compounds with a syntactically related term (in this case ‘ग’) as long as the compound does not end in a तिङ् affix.

In the compound, ‘आप + टा’ is placed in the prior position as per 2-2-30 उपसर्जनं पूर्वम्‌ – In a compound a term which has the designation ‘उपसर्जन’ should be placed in the prior position. Note: Here ‘आप + टा’ is the उपपदम् and hence it gets the उपसर्जन-सञ्ज्ञा by 1-2-43 प्रथमानिर्दिष्टं समास उपसर्जनम् – A term denoted by a word (in this case ‘उपपदम्’) ending in a nominative case in a सूत्रम् (in this case 2-2-19 उपपदमतिङ्) which prescribes a compound gets the designation ‘उपसर्जन’।

‘आप + टा + ग’ gets प्रातिपदिक-सञ्ज्ञा by 1-2-46 कृत्तद्धितसमासाश्च। This allows 2-4-71 सुपो धातुप्रातिपदिकयोः to apply in the next step.

(8) आप + ग । By 2-4-71 सुपो धातुप्रातिपदिकयोः – A सुँप् affix takes a लुक् elision when it is a part of a धातुः or a प्रातिपदिकम्।

= आपग ।

The प्रातिपदिकम् ‘आपग’ is an adjective. In the present example, it refers to the feminine noun ‘नदी’। Hence we form the feminine प्रातिपदिकम् ‘आपगा’ by adding the feminine affix टाप् as follows –

(9) आपग + टाप् । By 4-1-4 अजाद्यतष्टाप्‌ – The प्रातिपदिकानि ‘अज’ etc. and प्रातिपदिकानि ending in the letter ‘अ’ get the टाप् affix in the feminine gender.

(10) आपग + आ । अनुबन्ध-लोपः by 1-3-3 हलन्त्यम्, 1-3-7 चुटू, 1-3-9 तस्य लोपः।

(11) आपगा । By 6-1-101 अकः सवर्णे दीर्घः।

The विवक्षा is प्रथमा-बहुवचनम्। Hence the form is आपगाः।