The Philippian brethren were Paul's joy and crown in the sense that they were a result of his work. Remember that a congregation was established in Philippi because of Paul's efforts (cf. Acts 16). He was joyful because of their conversion to Christ. There is a certain sense in which they would be his reward at the final judgment, if they remained faithful. As I Corinthians 3:14 teaches - "If anyone's work which he has built on endures, he will receive a reward."
Paul continued in Philippians 4:2,3 - "I implore Euodia and I implore Syntyche to be of the same mind in the Lord. And I urge you also, true companion, help these women who labored with me in the gospel, with Clement also, and the rest of my fellow workers, whose names are in the Book of Life." Paul begged these two women to do what was right. They needed to be united or "of the same mind". All Christians are to seek unity. Consider I Corinthians 1:10 on this point - "Now I plead with you, brethren, by the name of our Lord Jesus Christ, that you all speak the same thing, and that there by no divisions among you, but that you be perfectly joined together in the same mind and in the same judgment."
These women were not doing that. There was a division between them. It seems safe to assume that these two women were members of the church in Philippi who were having some sort of quarrel or disagreement. The details of such are not recorded, but surely the Christians there were familiar with it. Evidently, Euodia and Syntyche were at odds with each other and this was affecting the church in a negative way. Paul implored them to set aside their personal differences and be of the same mind. They needed to seek the good of the church above their own personal interests.
It would seem that this was not a doctrinal dispute or Paul would have simply rebuked the one in error and declared the truth. On this occasion, however, he begged them both to get along as children of God should. How sad it is that these women are remembered for their disagreement. At one time they were fellow workers with Paul (cf. Phil. 4:3)! Hopefully, they repented of their quarrel and became united once again.
Today, we need to continually be reminded to get along with our brothers and sisters in Christ! If we are not careful, it is easy to get upset with each other unnecessarily and develop grudges. This is not the Lord's way! It is easy to lose sight of the importance of working for the Lord when we get busy arguing with one another over matters of opinion. May we pursue unity based on faithfulness to God's word and love for each other. May we not elevate our opinions and seek to get our way. May we strive for humility and obedience before God so that we may be unified with one another. May we, like the unnamed disciple Paul referred to as "true companion", work to keep the peace among the body of Christ.
It is of interest to note that Paul mentions the "Book of Life" at the end of Philippians 4:3. This is the only place in the New Testament that mentions this book (besides Revelation). Those who were faithful in Philippi had their names written in this book. In ancient cities there was often a list kept in which the names of all those living in that city were recorded. When a person was born or moved there, their name was added to the list. When a person died or moved from there, their name was removed from the list. Spiritually speaking, God keeps a list of all of those who are alive. That is, He has a record of those who have been born again--those who have been raised to walk in newness of life (cf. Rom. 6:3,4; John 3:3-5). May we always endeavor to stand fast in the Lord so that our names will not be erased from the Book of Life!