The participle often agrees with the logical, and not with the grammatical, subject. The participle thus often agrees with the subject of the finite verb which the writer had in mind when he began the sentence, but for which he later substitutes another verb; or the participle may later be used as if in agreement with the subject of another finite verb than the one actually employed.
a. A participle in the nominative may belong to a finite verb requiring an oblique case. Thus, ἀποβλέψα_ς πρὸς τοῦτον τὸν στόλον . . ., ἔδοξέ μοι πάγκαλος εἶναι (= ἡγησάμην πάγκαλον εἶναι)
to put to death not merely those who were there but also all the Mytilenaeans, urging against them their revolt, etc.
b. Two or more substantives or pronouns with their participles may stand in partitive apposition ( cross981) to the logical subject. Thus, τὰ περὶ Πύλον ὑπ' ἀμφοτέρων κατὰ κράτος ἐπολεμεῖτο (= ἀμφότεροι ἐπολέμουν),
Ἀθηναῖοι μὲν . . . τὴν νῆσον περιπλέοντες . . ., Πελοποννήσιοι δὲ ἐν τῇ ἡπείρῳ στρατοπεδευόμενοι
λόγοι δ' ἐν ἀλλήλοισιν ἐρρόθουν κακοί, φύλαξ ἐλέγχων φύλακα
θαυμάζοντες ἄλλος ἄλλῳ ἔλεγεν
c. Without regard to the following construction, a participle may stand in the nominative. The use of the genitive absolute would here be proper, but would cause the main subject of the thought to occupy a subordinate position. Thus, ἐπιπεσὼν τῇ Φαρναβάζου στρατοπεδείᾳ, τῆς μὲν προφυλακῆς αὐτοῦ Μυ_σῶν ὄντων πολλοὶ ἔπεσον
N. The nominative participle is sometimes found in clauses without a finite verb, but only when some finite verb is to be supplied (cp. Ψ 546), as with εἰ, ἐά_ν, ὅταν (
d. Likewise a participle may stand in the accusative or (rarely) in the dative when the construction demands another case. Thus, σοὶ δὲ συγγνώμη (= συγγνώμη ἐστὶ σὲ)
λέγειν τάδ' ἐστί, μὴ πάσχουσαν ὡς ἐγὼ κακῶς
τὸ μὲν μεθ' ἑαυτοῦ στρατόπεδον ἔχοντι ἐν τῷ ἰσθμῷ ἐπιτηρεῖν τοὺς Ἀθηναίους
Herbert Weir Smyth [n.d.], A Greek Grammar for Colleges; Machine readable text [info] [word count] [Smyth].