Liam Adams is expected to appeal the decision to extradite him to Northern Ireland to face multiple charges of sexually abusing his daughter.
Mr Adams, the brother of Sinn Fein president Gerry Adams, was released on bail yesterday after the High Court in Dublin ruled he should be extradited.
Mr Adams, who was supported in court by another daughter Claire Smith and her friend, showed no emotion as sections of the 64-page judgment were read by Mr Justice John Edwards.
The PSNI wants to question him in relation to 18 alleged offences against Aine Tyrell, who has waived her right to anonymity.
Yesterday, Judge Edwards said that the Republic's High Court would not uphold the points of objection raised by Liam Adams in his fight against extradition and would make an order for his surrender to authorities in Northern Ireland. He said issues relating to the possible prejudice to his trial on the grounds of pre-trial publicity, and the delay in bringing the prosecution, could be addressed by the courts here.
The judge said there was no evidence Mr Adams's prosecution was politically motivated.
People on both sides of the political spectrum in Northern Ireland, he said, including Liam Adams's "prominent brother", may have made politically motivated public pronouncements or commentated inappropriately in a self-serving way on the case.
But he said there was no evidence that the prosecution service in Northern Ireland had been subjected to or had yielded to pressure from external sources and acted out of political motivation.
The judge said that despite opinions as to Mr Adams's guilt expressed by prominent people, including his brother, it was quite incorrect for Mr Adams to contend that he no longer enjoyed the presumption of innocence.
But Judge Edwards deferred making the order for extradition until Thursday.
This will allow time for Mr Adams and his legal team to consider whether to appeal the judge's ruling.
Mr Adams, who denies all the charges, will remain on bail until Thursday's hearing and could remain on bail for up to two years if his case is appealed to the Supreme Court.
He is wanted in Northern Ireland to answer five charges of rape, seven charges of indecent assault and six charges of gross indecency against Ms Tyrell (38).
Speaking briefly as she left the court building, Ms Tyrell, who had travelled from Belfast for the hearing, said that she was feeling "more positive" after the ruling.
Flanked by family members, she said that she wished to thank her friends and family for supporting her, adding that she would be there "until the end".
"I'm feeling a bit more positive and I'm ready to take what comes," she said.
Story so far
The sex abuse claims against Liam Adams became public in December 2009 when his daughter Aine Tyrell was featured in a TV documentary. It is claimed 18 alleged sexual offences occurred in Belfast between March 1977 and March 1983 when the alleged victim was between four and 10 years old. Mr Adams turned himself in to gardai in March 2010 after a European Arrest Warrant was issued by the PSNI. He denies the charges.