The circumstantial participle is added, without the article, to a noun or pronoun to set forth some circumstance under which an action, generally the main action, takes place.
a. The circumstantial participle thus qualifies the principal verb of the sentence like an adverbial clause or supplementary predicate. Cp. μετὰ ταῦτα εἶπε
afterwards he said with γελῶν εἶπε
b. The circumstantial participle has no article. In agreement with a noun and its article, it stands before the article or after the noun (i.e. in the predicate position). By the agreement of the participle with a noun or pronoun, the predicate of the sentence is more exactly defined.
Herbert Weir Smyth [n.d.], A Greek Grammar for Colleges; Machine readable text [info] [word count] [Smyth].