He saw that Adam had on his shoulder a mysterious box, which he took to contain something valuable.

Seeing that Lady Arabella was secretly following Adam, he was confirmed in this idea.

His mind--such as it was--was fixed on her trying to steal, and he credited her at once with making use of this new opportunity.
In his walk, Adam went into the grounds of Castra Regis, and Oolanga saw her follow him with great secrecy.

He feared to go closer, as now on both sides of him were enemies who might make discovery.

When he realised that Lady Arabella was bound for the Castle, he devoted himself to following her with singleness of purpose.