The 119th Psalm is, as I’m sure you know, the longest in the Psalter by far. It’s so long that it has (probably?) never been appointed to be sung or said all the way through in a liturgical setting. Private recitation and devotion, is another matter. Thomas Cranmer’s monthly cycle of Psalms splits it over a few days, starting on the evening of the 24th day.
As you will find in most Bibles, Psalm 119 has 22 sections. These sections are noted in the Prayer Books also; four are grouped together in the evenings and five are grouped together in the mornings. These groups come from the structure of the original Hebrew poetry: an acrostic. The acrostic is actually a fairly common poetic structure in the Hebrew Bible: it’s a simple matter of beginning each successive line with the next letter of the alphabet. A handful of Psalms are acrostics, each chapter of the book of Lamentations is a sort of acrostic (well, chapter five is an anti-acrostic, but we’ll check that out later), and the occasional bit of prophetic writing also uses this device. Psalm 119, however, does this to the extreme: it has eight lines (verses) beginning with the first letter (aleph), then eight beginning with the second letter (beth), and so on, all the way through the alphabet. Obviously this effect is lost in translation, but many Bibles (and most if not all Prayer Books) note these eight-verse groupings.
One result of the acrostic structure is that the Psalm doesn’t have another organizing principle or logical flow. It’s a series of meditations on God’s law and commands (etc.), with little sense of progression from one section to the next. In that regard it’s like some of our modern songs (Christian or otherwise), dwelling on ideas, topics or feelings, but not developing a logical structure for the lyrics. This means that, in the context of the liturgy, we can fruitfully deal with each section of Psalm 119 as if it were its own psalm, without missing much context.
In medieval and early Prayer Book tradition, therefore, it was appointed that the worshiping congregation place the Glory be to the Father at the end of each section of Psalm 119. Today, Prayer Books tend to be ambiguous – we can either say that end the end of the whole Psalmody section of the Daily Office or at the end of each Psalm. But be it known here that if you opt for the latter option, which was the way of the early Prayer Books, you may even do so with each eight-verse section of Psalm 119.