The battle formation of the Sumerian phalanx, which is carved in relief on the reverse of the Stele of the Vultures, is studied in this article in light of textual sources, i. e., administrative documents, almost contemporary to the Stele, and later royal hymns and lexical texts.
DP 135 (dated around 2370 B. C.) suggests that a complete Sumerian phalanx was composed of thirty-two soldiers: a commander, a sub-commander, twenty-four spearmen and six shield-bearers. According to the Stele of the Vultures (produced 50 years or more earlier than DP 135), on the other hand, Lagashite troops of thirty men in phalanx formation, being led by King Eannatum, charged on the enemy. In my view, the phalanx of the Stele is composed of the following soldiers: a sub-commander, a shield-bearer who protects the sub-commander, twenty-four spearmen standing in four lines, and four shield-bearers protecting the first four spearmen of the respective lines. Although neither a commander nor a man holding a protective shield for the commander is found in the Stele, the battle formation drawn on the Stele differs only superficially from what DP 135 suggests. King Eannatum, whom the Stele depicts as standing unguarded in front of his troops in phalanx, plays the role of commander.
I reconstruct the phalanx of the Stele as follows. The head and the two feet that are found on the far right (as one faces it) in the phalanx relief of the Stele (called Head 1-Feet 1-2 in this article) represent the sub-commander of the troops. The second soldier, with the head and feet second from the right in relief (Head 2-Feet 3-4), protects the sub-commander with his large shield. The spearman of Head 3-Feet 5-6 stands at the head of the six men in the first line, being guarded by the shield-bearer of Head 4-Feet 7-8, and so on. On the far left of the Stele, the head and feet of the spearman, who is positioned first in the fourth line, are only imperfectly carved (Head 9-Feet 17-18). Two more heads are in relief on the left side board of the Stele, with the carvings of their four feet being completely lost from the board (Head 1′-Feet 1′-2′ Head 2′-Feet 3′-4′). I conclude that the spearman of Head 9-Feet 17-18, found last on the reverse, occurs again on the right of the side board (Head 1′-Feet 1′-2′) and that he is protected by the fifth shield held by the man of Head 2′-Feet 3′-4′ on his left.
The term ama-ERIN₂ refers to shield-bearers in DP 135. In the other contemporary text (Nik 1 3), however, it occurs as a designation of the whole army (composed of both shield-bearers and spearmen). The expression ama-erin₂-na of the later periods, which is often interpreted by Assyriologists to denote “the main body of the troops” on the basis of its Akkadian translation, seems to have been derived from ama-ERIN₂ with a meaning as found in Nik 1 3. Like Eannatum of Pre-Sargonic Lagash, King Shulgi of Ur, dated to the first half of the 21st century B. C., was obligated to stand alone before his troops (called ama-erin₂-na) [Šulgi B 31; Šulgi E 209]. It is rather doubtful, however, that Shulgi's troops were in phalanx formation as had been the case of the soldiers of Eannatum.
ERIN₂-suh₅-ha, which is used as a designation for spearmen in DP 135, occurs again in a later lexical text in slightly different writing (erin₂-suh). ERIN₂-suh₅-ha is possibly interpreted to mean “selected troops (of spearmen guarded by shield-holders).”

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地層の逆転がない露頭構造と層序の数学的表現として，構造グラフG＝（V，R^#，φ_V，φ_A）と層序グラフS＝（V/E^*，U^*，φ_V，φ_A）を提案してきた．今回，構造グラフGと層序グラフSを逆転のある露頭構造と層序の表現に拡張した．地質体間の接触関係をxR^#yと一般表現する．R^#＝R∨I∨F∨L∨D∨Tであり，xRyは空間的な上下関係，xIyは貫入，xFyは断層による切断，xLyは包含関係，xDyは同時異相，xTyは逆転の関係である．接触関係の名称の集合はA＝｛整合，不整合，貫入，断層，包含，同時異相，整合（逆転），不整合（逆転）｝である． 逆転関係の場合，新旧関係はxTy ⇒ yU*x，接触関係の名称はφ_A（x, y）＝不整合（逆転）⇒ φ_A（y, x）＝不整合，φ_A（x, y）＝整合（逆転）⇒ φ_A（y, x）＝整合と推論する． 複数の地質体［g_i］が逆転しており，これを複数の地質体［g_j ］が不整合に覆う場合，不整合の空間的に下側の地質体をH₁，上側の地質体をH₂ とすると，層序P’はP’＝（H₁，不整合，H₂ ）と表現される．H₁，H₂ の層序をそれぞれP_’1，P’₂ とすると，全体の層序は，P’＝（P’₁，不整合，P’₂ ）と推論された．

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露頭データベースの作成には層序と露頭構造の数学的表現が必要である．露頭構造と層序はグラフ理論と二項関係を使用した構造グラフと層序グラフで数学的に表現できる．逆転のない地層で構成される地質構造を対象とした構造グラフG＝（V, R）と層序グラフS＝（V, U^＊）を，火成岩の貫入や断層による切断がある露頭構造と層序の表現に拡張した．ここで，Vは地質体の集合，Rは接触関係，U^＊は新旧関係Uの反射的かつ推移的閉包を表す． 露頭で観察される２つの地質体xとyの空間的な接触関係を二項関係xR⁺yで一般表現することにより，貫入や断層が存在する露頭構造を構造グラフ（V, R⁺）として表現できた．ここで，R⁺ ＝｛（x, y）| xRy∨xIy∨xFy, x∈V, y∈V ｝であり，xRyは空間的な上下関係，xIyは貫入，xFyは断層による切断である．接触関係の情報はφA（x, y）＝整合のように弧のラベルで表現する． 地層累重の法則により上下関係xRyから新旧関係xUyが推論できることと，交差切りの法則により貫入関係xIyや断層関係xFyから新旧関係xUyが推論できることを組み合わせると，接触関係xR⁺yから新旧関係xUyが推論できるという一般規則が得られる．この推論規則で得られた新旧関係を使うことにより，貫入や断層を含む地質体の形成順序と接触関係を層序グラフ（V, U^＊）やラベル付きハッセ図で表現できた． 以上のようにして，火成岩の貫入や断層による切断がある地質構造に対する露頭構造や層序の表現が可能になった．

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