Система управления с позиции целенаправленного действия

ȼɟɫɬɧɢɤ ɆȽɌɍ, ɬɨɦ 3, ʋ1, 2000 ɝ.
ɫɬɪ.7-12
ɋɢɫɬɟɦɚ ɭɩɪɚɜɥɟɧɢɹ ɫ ɩɨɡɢɰɢɢ ɰɟɥɟɧɚɩɪɚɜɥɟɧɧɨɝɨ ɞɟɣɫɬɜɢɹ
ȼ.ɂ. Ɇɟɧɶɲɢɤɨɜ1, Ⱥ.ɇ. Ⱥɧɢɫɢɦɨɜ2, Ɏɭɪɝɚɫɚ Ⱦɟɫɚɥɟɧɶ Ɇɟɪɞɚɫɚ1
1
ɋɭɞɨɜɨɞɢɬɟɥɶɫɤɢɣ ɮɚɤɭɥɶɬɟɬ ɆȽɌɍ, ɤɚɮɟɞɪɚ ɫɭɞɨɜɨɠɞɟɧɢɹ
2
ɋɭɞɨɜɨɞɢɬɟɥɶɫɤɢɣ ɮɚɤɭɥɶɬɟɬ ɆȽɌɍ, ɤɚɮɟɞɪɚ ɭɩɪɚɜɥɟɧɢɹ ɫɭɞɧɨɦ
Ⱥɧɧɨɬɚɰɢɹ. ɋɬɚɬɶɹ ɪɚɫɫɦɚɬɪɢɜɚɟɬ ɩɪɨɛɥɟɦɵ ɭɩɪɚɜɥɟɧɢɹ ɛɟɡɨɩɚɫɧɨɫɬɶɸ ɦɨɪɫɤɨɝɨ ɫɭɞɨɯɨɞɫɬɜɚ ɢɥɢ
ɤɚɱɟɫɬɜɨɦ ɬɪɚɧɫɩɨɪɬɧɵɯ ɭɫɥɭɝ ɫ ɬɨɱɤɢ ɡɪɟɧɢɹ ɫɢɫɬɟɦɧɨɝɨ ɩɨɞɯɨɞɚ. ȼ ɫɜɟɬɟ ɩɪɢɧɹɬɢɹ Ɇɟɠɞɭɧɚɪɨɞɧɨɣ
Ɇɨɪɫɤɨɣ Ɉɪɝɚɧɢɡɚɰɢɟɣ (ɂɆɈ) Ɇɟɠɞɭɧɚɪɨɞɧɨɝɨ Ʉɨɞɟɤɫɚ ɩɨ ɭɩɪɚɜɥɟɧɢɸ ɛɟɡɨɩɚɫɧɨɣ ɷɤɫɩɥɭɚɬɚɰɢɟɣ
ɫɭɞɨɜ ɢ ɩɪɟɞɨɬɜɪɚɳɟɧɢɟɦ ɡɚɝɪɹɡɧɟɧɢɹ (ɆɄɍȻ) ɩɨɤɚɡɚɧɨ, ɱɬɨ ɩɨɞɯɨɞ ɤ ɨɩɪɟɞɟɥɟɧɢɸ ɫɢɫɬɟɦɵ
ɭɩɪɚɜɥɟɧɢɹ ɫ ɩɨɡɢɰɢɢ ɰɟɥɟɧɚɩɪɚɜɥɟɧɧɨɝɨ ɞɟɣɫɬɜɢɹ ɩɨɡɜɨɥɹɟɬ ɪɚɫɫɦɚɬɪɢɜɚɬɶ ɬɟɤɫɬ ɫɬɚɧɞɚɪɬɨɜ ɤɚɤ
ɚɤɫɢɨɦɚɬɢɱɟɫɤɭɸ ɝɪɭɩɩɭ ɫɭɠɞɟɧɢɣ.
Abstract. The paper considers problems of sea navigation safety management or transport services quality in the
terms of a system approach. It has been shown in the paper that the approach to a management system as
a teleological action allows to consider the safety management standards text as an axiomatic judgements group.
1. ȼɜɟɞɟɧɢɟ
ɉɨɜɵɲɟɧɢɟ ɤɨɧɤɭɪɟɧɬɨɫɩɨɫɨɛɧɨɫɬɢ ɞɟɹɬɟɥɶɧɨɫɬɢ ɨɬɟɱɟɫɬɜɟɧɧɵɯ ɫɭɞɨɜ ɧɚ ɪɵɧɤɟ
ɬɪɚɧɫɩɨɪɬɧɵɯ ɭɫɥɭɝ ɧɟɪɚɡɪɵɜɧɨ ɫɜɹɡɚɧɨ ɫ ɜɧɟɞɪɟɧɢɟɦ ɫɭɞɨɯɨɞɧɵɦɢ ɤɨɦɩɚɧɢɹɦɢ ɜ ɫɜɨɸ
ɨɪɝɚɧɢɡɚɰɢɨɧɧɭɸ ɞɟɹɬɟɥɶɧɨɫɬɶ ɞɜɭɯ ɦɟɠɞɭɧɚɪɨɞɧɨ-ɩɪɢɡɧɚɧɧɵɯ ɫɬɚɧɞɚɪɬɨɜ. ɉɟɪɜɵɦ ɢɡ ɧɢɯ
ɹɜɥɹɟɬɫɹ International Safety Management (ISM) Code, ɜɟɞɟɧɧɵɣ ɜ ɞɟɣɫɬɜɢɟ ɞɟɜɹɬɨɣ ɝɥɚɜɨɣ
Ɇɟɠɞɭɧɚɪɨɞɧɨɣ ɤɨɧɜɟɧɰɢɢ "ɋɨɥɚɫ-74" ɫ ɩɨɩɪɚɜɤɚɦɢ ɢ ɞɨɩɨɥɧɟɧɢɹɦɢ. Ʉɨɞɟɤɫ ɢɦɟɟɬ ɨɛɹɡɚɬɟɥɶɧɵɣ
ɯɚɪɚɤɬɟɪ ɢ ɜ ɩɟɪɜɭɸ ɨɱɟɪɟɞɶ ɨɩɪɟɞɟɥɹɟɬ ɝɚɪɚɧɬɢɪɨɜɚɧɧɵɣ ɭɪɨɜɟɧɶ ɛɟɡɨɩɚɫɧɨɫɬɢ ɦɨɪɫɤɨɝɨ
ɫɭɞɨɯɨɞɫɬɜɚ, ɩɪɢ ɤɨɬɨɪɨɦ ɫɭɞɨɯɨɞɧɚɹ ɤɨɦɩɚɧɢɹ ɩɪɟɞɫɬɚɜɥɹɟɬ ɩɨɬɪɟɛɢɬɟɥɹɦ ɫɜɨɢ ɬɪɚɧɫɩɨɪɬɧɵɟ
ɭɫɥɭɝɢ. Ɇɨɠɧɨ ɩɨ-ɪɚɡɧɨɦɭ ɨɬɧɨɫɢɬɶɫɹ ɤ ɩɨɥɢɬɢɱɟɫɤɢɦ ɢ ɷɤɨɧɨɦɢɱɟɫɤɢɦ ɚɫɩɟɤɬɚɦ ɜɧɟɞɪɟɧɢɹ ISM
Code. Ɉɞɧɚɤɨ ɜ ɥɸɛɨɦ ɫɥɭɱɚɟ ɧɟɥɶɡɹ ɧɟ ɩɪɢɡɧɚɬɶ, ɱɬɨ ɜɜɟɞɟɧɢɟ ɜ ɞɟɣɫɬɜɢɟ Ʉɨɞɟɤɫɚ ɫɩɨɫɨɛɧɨ ɪɟɲɢɬɶ
ɩɪɨɛɥɟɦɭ ɜɵɜɨɞɚ ɢɡ ɷɤɫɩɥɭɚɬɚɰɢɢ "ɫɬɚɪɵɯ" ɫɭɞɨɜ. Ɇɟɬɨɞɚɦɢ ɡɚɩɪɟɬɚ, ɬɚɤɢɦɢ, ɧɚɩɪɢɦɟɪ, ɤɨɬɨɪɵɦɢ
ɩɨɥɶɡɭɸɬɫɹ ɜɥɚɫɬɢ ɋɒȺ, ɦɨɠɧɨ ɞɨɫɬɢɱɶ ɠɟɥɚɟɦɵɯ ɪɟɡɭɥɶɬɚɬɨɜ, ɧɨ ɷɬɢ ɪɟɡɭɥɶɬɚɬɵ ɥɨɤɚɥɶɧɵ ɢ
ɞɟɣɫɬɜɭɸɬ ɥɢɲɶ ɜ ɩɪɟɞɟɥɚɯ ɸɪɢɫɞɢɤɰɢɢ ɩɪɢɛɪɟɠɧɨɝɨ ɝɨɫɭɞɚɪɫɬɜɚ. ȼ ɦɟɠɞɭɧɚɪɨɞɧɨɦ ɩɥɚɧɟ ɫɥɟɞɭɟɬ
ɪɭɤɨɜɨɞɫɬɜɨɜɚɬɶɫɹ ɛɨɥɟɟ ɨɛɳɢɦɢ ɷɤɨɧɨɦɢɱɟɫɤɢɦɢ ɪɵɱɚɝɚɦɢ. ISM Code ɢ ɹɜɥɹɟɬɫɹ ɬɟɦ
ɷɤɨɧɨɦɢɱɟɫɤɢɦ ɪɵɱɚɝɨɦ, ɤɨɬɨɪɵɣ ɞɟɥɚɟɬ ɷɤɫɩɥɭɚɬɚɰɢɸ ɫɬɚɪɵɯ ɫɭɞɨɜ ɧɟɪɟɧɬɚɛɟɥɶɧɨɣ, ɩɨɫɤɨɥɶɤɭ
ɡɚɬɪɚɬɵ, ɧɚɩɪɚɜɥɹɟɦɵɟ ɧɚ ɩɨɞɞɟɪɠɚɧɢɟ ɧɭɠɧɨɝɨ ɭɪɨɜɧɹ ɛɟɡɨɩɚɫɧɨɫɬɢ ɫɭɞɨɜ, ɦɨɝɭɬ ɩɪɟɜɵɲɚɬɶ ɞɨɯɨɞ
ɨɬ ɢɯ ɩɪɨɢɡɜɨɞɫɬɜɟɧɧɨɣ ɞɟɹɬɟɥɶɧɨɫɬɢ.
Ɉɛɹɡɚɬɟɥɶɧɵɣ ɯɚɪɚɤɬɟɪ ɢɫɩɨɥɧɟɧɢɹ ɩɨɥɨɠɟɧɢɣ ISM Code ɧɟ ɫɩɨɫɨɛɟɧ ɜɥɢɹɬɶ ɧɚ ɤɚɱɟɫɬɜɨ
ɬɪɚɧɫɩɨɪɬɧɵɯ ɭɫɥɭɝ, ɚ ɫɨɡɞɚɟɬ ɬɨɥɶɤɨ ɭɫɥɨɜɢɹ ɞɥɹ ɢɯ ɝɚɪɚɧɬɢɪɨɜɚɧɧɨɝɨ ɩɪɟɞɨɫɬɚɜɥɟɧɢɹ ɡɚɤɚɡɱɢɤɭ ɫ
ɦɢɧɢɦɢɡɚɰɢɟɣ ɪɢɫɤɚ ɞɥɹ ɫɭɞɨɜɨɝɨ ɷɤɢɩɚɠɚ, ɝɪɭɡɚ ɢ ɷɤɨɥɨɝɢɢ ɦɨɪɫɤɨɝɨ ɩɪɨɫɬɪɚɧɫɬɜɚ. Ɉɬɫɸɞɚ
ɧɚɩɪɚɲɢɜɚɟɬɫɹ ɞɨɫɬɚɬɨɱɧɨ ɩɪɨɫɬɨɣ ɜɵɜɨɞ ɨ ɬɨɦ, ɱɬɨ ɩɨɫɬɪɨɟɧɢɟ ɢ ɷɤɫɩɥɭɚɬɚɰɢɹ ɫɢɫɬɟɦɵ ɭɩɪɚɜɥɟɧɢɹ
ɛɟɡɨɩɚɫɧɨɫɬɶɸ ɜɧɭɬɪɢ ɨɪɝɚɧɢɡɚɰɢɨɧɧɨɣ ɫɬɪɭɤɬɭɪɵ ɫɭɞɨɯɨɞɧɨɣ ɤɨɦɩɚɧɢɢ ɞɨɥɠɧɵ ɜ ɛɨɥɶɲɟɣ ɫɜɨɟɣ
ɱɚɫɬɢ ɧɨɫɢɬɶ ɡɚɬɪɚɬɧɵɣ ɯɚɪɚɤɬɟɪ. Ⱥɞɦɢɧɢɫɬɪɚɰɢɹɦ ɫɭɞɨɯɨɞɧɵɯ ɤɨɦɩɚɧɢɣ ɫɥɟɞɭɟɬ ɫɦɢɪɢɬɶɫɹ ɫ
ɡɚɬɪɚɬɚɦɢ ɨɬ ɷɤɫɩɥɭɚɬɚɰɢɢ ɫɢɫɬɟɦɵ ɭɩɪɚɜɥɟɧɢɹ ɛɟɡɨɩɚɫɧɨɫɬɶɸ (ɋɍȻ) ɢ ɢɫɤɚɬɶ ɩɭɬɢ ɩɨɜɵɲɟɧɢɹ
ɷɮɮɟɤɬɢɜɧɨɫɬɢ ɞɟɹɬɟɥɶɧɨɫɬɢ ɧɚ ɪɵɧɤɟ ɬɪɚɧɫɩɨɪɬɧɵɯ ɭɫɥɭɝ ɡɚ ɫɱɟɬ ɭɩɪɚɜɥɟɧɢɹ ɤɚɱɟɫɬɜɨɦ ɷɬɢɯ ɭɫɥɭɝ.
Ɉɛɳɢɦ ɧɚɩɪɚɜɥɟɧɢɟɦ ɩɪɢ ɭɩɪɚɜɥɟɧɢɢ ɤɚɱɟɫɬɜɨɦ ɬɪɚɧɫɩɨɪɬɧɵɯ ɭɫɥɭɝ ɹɜɥɹɟɬɫɹ ɞɨɫɬɚɬɨɱɧɨ
ɯɨɪɨɲɨ ɢɡɜɟɫɬɧɵɣ ɩɪɢɧɰɢɩ: "ɡɚɤɚɡɱɢɤ ɜɫɟɝɞɚ ɩɪɚɜ". Ⱦɨɤɭɦɟɧɬɨɦ, ɤɨɬɨɪɵɦ ɞɨɥɠɧɚ
ɪɭɤɨɜɨɞɫɬɜɨɜɚɬɶɫɹ ɫɭɞɨɯɨɞɧɚɹ ɤɨɦɩɚɧɢɹ, ɨɫɭɳɟɫɬɜɥɹɹ ɷɤɫɩɚɧɫɢɸ ɧɚ ɪɵɧɤɟ ɬɪɚɧɫɩɨɪɬɧɵɯ ɭɫɥɭɝ
ɞɨɥɠɟɧ ɛɵɬɶ ɨɞɢɧ ɢɡ ɫɟɪɢɢ ɫɬɚɧɞɚɪɬɨɜ ISO 9000. Ɉɪɝɚɧɢɡɚɰɢɨɧɧɚɹ ɫɬɪɭɤɬɭɪɚ, ɨɬɜɟɱɚɸɳɚɹ ɨɛɳɢɦ
ɩɪɢɧɰɢɩɚɦ ɫɬɚɧɞɚɪɬɨɜ ISO, ɭɩɪɚɜɥɹɹ ɤɚɱɟɫɬɜɨɦ ɭɫɥɭɝ, ɫɩɨɫɨɛɧɚ ɫɨɡɞɚɜɚɬɶ ɭɫɬɨɣɱɢɜɵɟ ɮɨɪɦɵ
ɜɡɚɢɦɨɫɜɹɡɢ ɫ ɡɚɤɚɡɱɢɤɨɦ, ɩɪɨɡɪɚɱɧɨɫɬɶ ɷɬɨɣ ɜɡɚɢɦɨɫɜɹɡɢ ɢ, ɡɚ ɫɱɟɬ ɢɧɬɟɧɫɢɮɢɤɚɰɢɢ ɷɤɫɩɥɭɚɬɚɰɢɢ
ɫɜɨɟɝɨ ɮɥɨɬɚ, ɩɨɥɭɱɚɬɶ ɞɨɩɨɥɧɢɬɟɥɶɧɭɸ ɩɪɢɛɵɥɶ. ȿɫɬɟɫɬɜɟɧɧɨ, ɱɬɨ ɱɚɫɬɶ ɞɨɩɨɥɧɢɬɟɥɶɧɨɣ ɩɪɢɛɵɥɢ,
ɩɨɥɭɱɟɧɧɨɣ ɨɬ ɷɤɫɩɥɭɚɬɚɰɢɢ ɫɢɫɬɟɦɵ ɭɩɪɚɜɥɟɧɢɹ ɤɚɱɟɫɬɜɨɦ, ɫɭɞɨɯɨɞɧɚɹ ɤɨɦɩɚɧɢɹ ɨɛɹɡɚɧɚ
ɧɚɩɪɚɜɢɬɶ ɧɚ ɩɨɞɞɟɪɠɚɧɢɟ ɢ ɫɨɜɟɪɲɟɧɫɬɜɨɜɚɧɢɟ ɫɜɨɟɣ ɫɢɫɬɟɦɵ ɩɨ ɭɩɪɚɜɥɟɧɢɸ ɛɟɡɨɩɚɫɧɨɫɬɶɸ.
ȼɡɚɢɦɨɡɚɜɢɫɢɦɨɫɬɶ ɫɢɫɬɟɦ ɭɩɪɚɜɥɟɧɢɹ ɛɟɡɨɩɚɫɧɨɫɬɶɸ ɢ ɤɚɱɟɫɬɜɨɦ ɬɪɚɧɫɩɨɪɬɧɵɯ ɭɫɥɭɝ ɹɜɥɹɟɬɫɹ
ɨɛɹɡɚɬɟɥɶɧɵɦ ɦɨɪɮɨɥɨɝɢɱɟɫɤɢɦ ɩɪɢɡɧɚɤɨɦ ɨɛɳɟɣ ɨɪɝɚɧɢɡɚɰɢɨɧɧɨɣ ɫɬɪɭɤɬɭɪɵ ɫɭɞɨɯɨɞɧɨɣ
ɤɨɦɩɚɧɢɢ. Ɍɚɤ, ɥɸɛɨɣ ɫɭɳɟɫɬɜɟɧɧɵɣ ɫɛɨɣ ɜ ɪɚɛɨɬɟ ɋɍȻ ɩɪɢɜɨɞɢɬ ɤ ɫɧɢɠɟɧɢɸ ɝɚɪɚɧɬɢɣ ɜ
7
Ɇɟɧɶɲɢɤɨɜ ȼ.ɂ. ɢ ɞɪ. ɋɢɫɬɟɦɚ ɭɩɪɚɜɥɟɧɢɹ ɫ ɩɨɡɢɰɢɢ ɰɟɥɟɧɚɩɪɚɜɥɟɧɧɨɝɨ ɞɟɣɫɬɜɢɹ
ɩɪɟɞɨɫɬɚɜɥɟɧɢɢ ɡɚɤɚɡɱɢɤɭ ɧɭɠɧɵɯ ɬɪɚɧɫɩɨɪɬɧɵɯ ɭɫɥɭɝ, ɱɬɨ ɧɟɢɡɛɟɠɧɨ ɫɤɚɠɟɬɫɹ ɧɚ ɭɪɨɜɧɟ
ɤɨɧɤɭɪɟɧɬɨɫɩɨɫɨɛɧɨɫɬɢ ɫɚɦɨɣ ɤɨɦɩɚɧɢɢ. ȼ ɬɨɠɟ ɜɪɟɦɹ ɨɲɢɛɤɢ ɦɟɧɟɞɠɦɟɧɬɚ ɫɩɨɫɨɛɧɵ ɫɧɢɡɢɬɶ
ɞɨɯɨɞɵ ɤɨɦɩɚɧɢɢ ɢ, ɫɥɟɞɨɜɚɬɟɥɶɧɨ, ɭɦɟɧɶɲɢɬɶ ɨɬɱɢɫɥɟɧɢɹ, ɧɚɩɪɚɜɥɹɟɦɵɟ ɧɚ ɩɨɞɞɟɪɠɚɧɢɟ ɧɭɠɧɨɝɨ
ɭɪɨɜɧɹ ɛɟɡɨɩɚɫɧɨɫɬɢ ɫɭɞɨɜ. ɉɨɷɬɨɦɭ ɜɨɡɧɢɤɚɟɬ ɧɟɨɛɯɨɞɢɦɨɫɬɶ ɜ ɢɫɫɥɟɞɨɜɚɧɢɢ ɞɟɹɬɟɥɶɧɨɫɬɢ ɫɢɫɬɟɦ
ɭɩɪɚɜɥɟɧɢɹ ɛɟɡɨɩɚɫɧɨɫɬɶɸ ɢ ɤɚɱɟɫɬɜɨɦ ɭɫɥɭɝ, ɤɨɬɨɪɵɟ ɫɨɡɞɚɸɬɫɹ ɢ ɮɭɧɤɰɢɨɧɢɪɭɸɬ ɜ ɪɚɦɤɚɯ
ɦɟɠɞɭɧɚɪɨɞɧɵɯ ɫɬɚɧɞɚɪɬɨɜ, ɚ ɬɚɤɠɟ ɩɪɨɰɟɫɫɨɜ ɜɡɚɢɦɨɞɟɣɫɬɜɢɹ ɦɟɠɞɭ ɧɢɦɢ.
ȼ ɫɩɟɰɢɚɥɶɧɨɣ ɥɢɬɟɪɚɬɭɪɟ, ɢɡɞɚɜɚɟɦɨɣ Ɋɨɫɫɢɣɫɤɢɦ Ɇɨɪɫɤɢɦ Ɋɟɝɢɫɬɪɨɦ ɢ ɆȺɄɈ, ɩɪɢɜɨɞɢɬɫɹ
ɞɨɫɬɚɬɨɱɧɨ ɛɨɥɶɲɨɣ ɫɩɢɫɨɤ ɩɪɢɡɧɚɤɨɜ ɛɥɢɡɨɫɬɢ ɦɟɠɞɭ ɫɬɚɧɞɚɪɬɚɦɢ ISM ɢ ISO. Ɍɚɤɚɹ ɞɟɬɚɥɢɡɚɰɢɹ
ɫɬɟɩɟɧɢ ɛɥɢɡɨɫɬɢ ɫɤɨɪɟɟ ɡɚɬɟɧɹɟɬ ɫɭɬɶ, ɧɨ ɧɟ ɞɚɟɬ ɨɬɜɟɬɚ ɧɚ ɜɨɩɪɨɫ ɨ ɜɡɚɢɦɨɫɜɹɡɢ ɫɬɚɧɞɚɪɬɨɜ. ȼ
ɞɟɣɫɬɜɢɬɟɥɶɧɨɫɬɢ ɫɥɟɞɭɟɬ ɨɝɪɚɧɢɱɢɬɶɫɹ ɥɢɲɶ ɨɞɧɢɦ ɩɪɢɡɧɚɤɨɦ, ɤɨɬɨɪɵɣ ɩɨ ɫɜɨɟɦɭ ɯɚɪɚɤɬɟɪɭ ɢɦɟɟɬ
ɩɪɢɧɰɢɩɢɚɥɶɧɨɟ ɡɧɚɱɟɧɢɟ. ȼɵɞɟɥɢɬɶ ɮɭɧɞɚɦɟɧɬɚɥɶɧɵɣ ɩɪɢɡɧɚɤ ɨɛɳɧɨɫɬɢ ɦɟɠɞɭ ɫɬɚɧɞɚɪɬɚɦɢ ɦɨɠɧɨ,
ɟɫɥɢ ɭɱɟɫɬɶ ɱɬɨ
ISM ŀ ISO
SM,
(1)
ɝɞɟ SM – ɨɛɳɧɨɫɬɶ ɫɬɚɧɞɚɪɬɨɜ, ɨɩɪɟɞɟɥɟɧɧɚɹ ɱɟɪɟɡ ɩɨɧɹɬɢɟ "ɫɢɫɬɟɦɚ ɭɩɪɚɜɥɟɧɢɹ".
ɋɢɫɬɟɦɵ ɭɩɪɚɜɥɟɧɢɹ ɜɢɞɚ (1) ɫɨɡɞɚɸɬɫɹ ɜ ɪɚɦɤɚɯ ɨɪɝɚɧɢɡɚɰɢɨɧɧɨɣ ɫɬɪɭɤɬɭɪɵ ɤɨɦɩɚɧɢɢ, ɤɚɤ
ɫɪɟɞɫɬɜɨ, ɨɛɟɫɩɟɱɢɜɚɸɳɟɟ ɩɪɨɜɟɞɟɧɢɟ ɨɩɪɟɞɟɥɟɧɧɨɣ ɩɨɥɢɬɢɤɢ ɜ ɞɨɫɬɢɠɟɧɢɢ ɩɨɫɬɚɜɥɟɧɧɵɯ ɰɟɥɟɣ ɜ
ɨɛɥɚɫɬɢ ɤɚɱɟɫɬɜɚ ɬɪɚɧɫɩɨɪɬɧɵɯ ɭɫɥɭɝ, ɥɨɤɚɥɢɡɨɜɚɧɧɵɯ ɬɪɟɛɨɜɚɧɢɹɦɢ ɩɨ ɨɛɟɫɩɟɱɟɧɢɸ ɛɟɡɨɩɚɫɧɨɫɬɢ
ɦɨɪɫɤɨɝɨ ɫɭɞɨɯɨɞɫɬɜɚ (ɉɚɥɟɯɨɜ, 1998). ɉɨɷɬɨɦɭ ɫɢɫɬɟɦɧɨɟ ɢɫɫɥɟɞɨɜɚɧɢɟ ɞɨɥɠɧɨ ɛɵɬɶ ɨɫɧɨɜɨɣ ɤɚɤ ɩɪɢ
ɩɨɫɬɪɨɟɧɢɢ, ɬɚɤ ɢ ɩɪɢ ɷɤɫɩɥɭɚɬɚɰɢɢ ɫɢɫɬɟɦ ɭɩɪɚɜɥɟɧɢɹ ɛɟɡɨɩɚɫɧɨɫɬɶɸ ɢ ɤɚɱɟɫɬɜɨɦ ɭɫɥɭɝ. Ȼɨɥɟɟ ɬɨɝɨ,
ɨɫɨɛɟɧɧɨɫɬɢ ɜ ɮɭɧɤɰɢɨɧɢɪɨɜɚɧɢɢ ɨɞɧɨɣ ɫɢɫɬɟɦɵ ɜ ɫɢɥɭ ɫɜɹɡɚɧɧɨɫɬɢ ɦɨɠɧɨ ɩɟɪɟɧɨɫɢɬɶ ɧɚ ɞɪɭɝɭɸ, ɯɨɬɹ
ɬɚɤɨɣ ɩɟɪɟɧɨɫ ɫɥɟɞɭɟɬ ɨɫɭɳɟɫɬɜɥɹɬɶ ɫ ɨɩɪɟɞɟɥɟɧɧɨɣ ɞɨɥɟɣ ɨɫɬɨɪɨɠɧɨɫɬɢ. ɋɢɫɬɟɦɧɵɣ ɩɨɞɯɨɞ ɢ
ɫɢɫɬɟɦɧɨɟ ɢɫɫɥɟɞɨɜɚɧɢɟ ɧɟ ɹɜɥɹɸɬɫɹ ɱɟɦ-ɬɨ ɩɪɢɧɰɢɩɢɚɥɶɧɨ ɧɨɜɵɦ, ɚ ɩɪɟɞɫɬɚɜɥɹɸɬ ɫɨɛɨɣ ɟɫɬɟɫɬɜɟɧɧɵɣ
ɢ, ɧɚɜɟɪɧɨɟ, ɟɞɢɧɫɬɜɟɧɧɵɣ ɧɚɭɱɧɵɣ ɦɟɬɨɞ ɪɟɲɟɧɢɹ ɬɟɨɪɟɬɢɱɟɫɤɢɯ ɢ ɩɪɚɤɬɢɱɟɫɤɢɯ ɡɚɞɚɱ, ɩɨɫɬɚɜɥɟɧɧɵɯ
ɩɟɪɟɞ ɫɭɞɨɜɥɚɞɟɥɶɰɚɦɢ, ɫ ɨɞɧɨɣ ɫɬɨɪɨɧɵ, ɫɚɦɢɦ ɪɵɧɤɨɦ ɬɪɚɧɫɩɨɪɬɧɵɯ ɭɫɥɭɝ, ɚ ɫ ɞɪɭɝɨɣ ɫɬɨɪɨɧɵ,
ɞɟɜɹɬɨɣ ɝɥɚɜɨɣ Ɇɟɠɞɭɧɚɪɨɞɧɨɣ Ʉɨɧɜɟɧɰɢɢ "ɋɨɥɚɫ-74" ɢ ɤɨɞɟɤɫɨɦ ISM ɤ ɧɟɣ.
ȼ ɚɪɫɟɧɚɥɟ ɨɛɳɟɝɨ ɫɢɫɬɟɦɧɨɝɨ ɢɫɫɥɟɞɨɜɚɧɢɹ ɢɦɟɟɬɫɹ (ɗɧɰɢɤɥɨɩɟɞɢɹ ɤɢɛɟɪɧɟɬɢɤɢ, 1974), ɜɨɩɟɪɜɵɯ, ɬɟɨɪɢɹ ɚɛɫɬɪɚɤɬɧɵɯ ɫɢɫɬɟɦ, ɚ ɜɨ-ɜɬɨɪɵɯ, ɬɟɨɪɢɹ ɤɨɧɤɪɟɬɧɵɯ ɫɢɫɬɟɦ, ɜɤɥɸɱɚɸɳɚɹ ɦɟɬɨɞɨɥɨɝɢɸ
ɚɧɚɥɢɡɚ ɢ ɫɢɧɬɟɡɚ. ɉɪɢ ɜɵɩɨɥɧɟɧɢɢ ɤɨɧɤɪɟɬɧɵɯ ɫɢɫɬɟɦɧɵɯ ɢɫɫɥɟɞɨɜɚɧɢɣ ɞɨɫɬɚɬɨɱɧɨ ɱɚɫɬɨ ɦɨɠɧɨ
ɜɫɬɪɟɬɢɬɶɫɹ ɫ ɩɨɩɵɬɤɚɦɢ ɞɚɬɶ ɩɪɢɨɪɢɬɟɬ ɢɥɢ ɬɟɨɪɢɢ ɚɛɫɬɪɚɤɬɧɵɯ ɫɢɫɬɟɦ, ɢɥɢ ɬɟɨɪɢɢ ɤɨɧɤɪɟɬɧɵɯ
ɫɢɫɬɟɦ. Ⱥɧɬɚɝɨɧɢɡɦ ɬɟɨɪɢɣ ɩɪɨɹɜɥɹɟɬɫɹ ɭɠɟ ɞɚɠɟ ɜ ɫɚɦɨɦ ɨɩɪɟɞɟɥɟɧɢɢ ɫɢɫɬɟɦɵ ɤɚɤ ɬɚɤɨɜɨɣ. Ɍɚɤ, ɜ
ɬɟɨɪɢɢ ɚɛɫɬɪɚɤɬɧɵɯ ɫɢɫɬɟɦ ɡɚ ɧɟɟ ɩɪɢɧɢɦɚɟɬɫɹ ɦɧɨɠɟɫɬɜɨ ɜɡɚɢɦɨɫɜɹɡɚɧɧɵɯ ɷɥɟɦɟɧɬɨɜ, ɤɨɬɨɪɵɟ ɩɨ
ɫɜɨɟɣ ɫɭɬɢ ɹɜɥɹɸɬɫɹ ɩɨɧɹɬɢɹɦɢ. Ɍɟɨɪɢɹ ɠɟ ɤɨɧɤɪɟɬɧɵɯ ɫɢɫɬɟɦ ɨɩɪɟɞɟɥɹɟɬ ɫɢɫɬɟɦɭ ɤɚɤ ɦɧɨɠɟɫɬɜɨ
ɜɡɚɢɦɨɫɜɹɡɚɧɧɵɯ ɷɥɟɦɟɧɬɨɜ, ɤɨɬɨɪɵɟ ɨɛɥɚɞɚɸɬ ɫɜɨɣɫɬɜɚɦɢ ɮɢɡɢɱɟɫɤɢɯ ɨɛɴɟɤɬɨɜ. ȼ ɬɨ ɠɟ ɜɪɟɦɹ
ɞɨɩɭɫɬɢɦɨ ɫɱɢɬɚɬɶ, ɱɬɨ ɬɟɨɪɢɹ ɭɩɪɚɜɥɹɟɦɵɯ ɫɢɫɬɟɦ ɦɨɠɟɬ ɢɦɟɬɶ ɟɞɢɧɭɸ ɨɛɳɭɸ ɢɫɯɨɞɧɭɸ ɬɨɱɤɭ,
ɨɩɪɟɞɟɥɹɟɦɭɸ ɫ ɩɨɦɨɳɶɸ ɩɨɧɹɬɢɹ "ɰɟɥɟɧɚɩɪɚɜɥɟɧɧɨɟ ɞɟɣɫɬɜɢɟ". Ƚɢɩɨɬɟɡɚ ɨ ɫɜɹɡɚɧɧɨɫɬɢ ɬɟɨɪɢɣ
ɚɛɫɬɪɚɤɬɧɵɯ ɢ ɤɨɧɤɪɟɬɧɵɯ ɭɩɪɚɜɥɹɟɦɵɯ ɫɢɫɬɟɦ ɩɨɡɜɨɥɢɬ ɜɜɟɫɬɢ ɜ ɫɢɫɬɟɦɧɨɟ ɢɫɫɥɟɞɨɜɚɧɢɟ ɚɛɫɬɪɚɤɬɧɵɯ
ɫɢɫɬɟɦ ɨɛɳɢɟ ɩɪɢɧɰɢɩɵ ɦɟɬɨɞɨɥɨɝɢɢ ɚɧɚɥɢɡɚ ɢ ɫɢɧɬɟɡɚ ɤɨɧɤɪɟɬɧɵɯ ɫɢɫɬɟɦ.
2. ɐɟɥɟɧɚɩɪɚɜɥɟɧɧɨɟ ɞɟɣɫɬɜɢɟ ɢ ɭɩɪɚɜɥɹɟɦɚɹ ɫɢɫɬɟɦɚ
Ɉɛɳɧɨɫɬɶ ɩɨɞɯɨɞɚ ɤ ɭɩɪɚɜɥɹɟɦɨɣ ɫɢɫɬɟɦɟ ɜ ɪɚɦɤɚɯ ɰɟɥɟɧɚɩɪɚɜɥɟɧɧɨɝɨ ɞɟɣɫɬɜɢɹ ɞɚɟɬ ɩɪɚɜɨ
ɪɚɫɫɦɚɬɪɢɜɚɬɶ ɟɟ ɤɚɤ ɨɬɨɛɪɚɠɟɧɢɟ ¦, ɩɨɫɬɪɨɟɧɧɨɟ ɧɚ ɨɛɴɟɞɢɧɟɧɢɢ ɞɜɭɯ ɮɭɧɤɰɢɣ: ɜɧɟɲɧɟɣ ɢ
ɜɧɭɬɪɟɧɧɟɣ. Ɍɚɤɨɟ ɨɛɴɟɞɢɧɟɧɢɟ ɫɥɟɞɭɟɬ ɨɩɪɟɞɟɥɢɬɶ, ɤɚɤ ɧɟɤɨɬɨɪɨɟ ɫɨɝɥɚɫɨɜɚɧɧɨɟ ɦɧɨɠɟɫɬɜɨ ɜɢɞɚ,
ɩɨɥɭɱɟɧɧɨɟ ɜ ɪɟɡɭɥɶɬɚɬɟ ɫɥɟɞɭɸɳɟɣ ɨɩɟɪɚɰɢɢ:
¦ : F1 F2 o L,
(2)
ɝɞɟ > F1 @  F1; > F2 @  F2, ɚ L – ɰɟɥɟɧɚɩɪɚɜɥɟɧɧɨɟ ɞɟɣɫɬɜɢɟ ɜ ɭɩɪɚɜɥɹɟɦɨɣ ɫɢɫɬɟɦɟ, ɨɬɜɟɱɚɸɳɟɟ
ɭɫɥɨɜɢɹɦ, ɡɚɤɪɟɩɥɟɧɧɵɦ ɜɨ ɜɧɟɲɧɟɣ ɢ ɜɧɭɬɪɟɧɧɟɣ ɮɭɧɤɰɢɹɯ.
ȼ ɫɨɝɥɚɫɨɜɚɧɧɨɦ ɦɧɨɠɟɫɬɜɟ (2) ɩɪɢɧɹɬɨ, ɱɬɨ ɜɧɟɲɧɹɹ ɮɭɧɤɰɢɹ F1 ɹɜɥɹɟɬɫɹ ɩɪɟɨɛɥɚɞɚɸɳɟɣ ɢ
ɫɥɭɠɢɬ ɩɪɢɱɢɧɨɣ, ɮɨɪɦɢɪɭɸɳɟɣ ɜɧɭɬɪɟɧɧɸɸ ɮɭɧɤɰɢɸ F2 ɭɩɪɚɜɥɹɟɦɨɣ ɫɢɫɬɟɦɵ, ɬ.ɟ.
F1 o F2.
8
(3)
ȼɟɫɬɧɢɤ ɆȽɌɍ, ɬɨɦ 3, ʋ1, 2000 ɝ.
ɫɬɪ.7-12
ȼ ɬɟɯɧɢɱɟɫɤɢɯ ɢ ɨɪɝɚɧɢɡɚɰɢɨɧɧɵɯ ɫɢɫɬɟɦɚɯ ɭɩɪɚɜɥɟɧɢɹ ɫɜɹɡɶ ɦɟɠɞɭ ɩɪɢɱɢɧɨɣ ɢ ɫɥɟɞɫɬɜɢɟɦ
ɜɢɞɚ (3) ɜɫɟɝɞɚ ɢɦɟɟɬ ɦɟɫɬɨ. ɂɦɟɧɧɨ ɜɧɟɲɧɹɹ ɮɭɧɤɰɢɹ ɫɩɨɫɨɛɧɚ ɩɨɞɜɟɫɬɢ ɤ ɩɪɟɞɫɬɚɜɥɟɧɢɸ ɨ
ɜɧɭɬɪɟɧɧɟɣ ɫɬɪɭɤɬɭɪɟ ɫɢɫɬɟɦɵ, ɚ ɬɚɤɠɟ ɞɚɬɶ ɜɨɡɦɨɠɧɨɫɬɶ ɨɰɟɧɢɬɶ ɤɚɱɟɫɬɜɨ ɭɩɪɚɜɥɟɧɢɹ. Ⱦɥɹ ɷɬɨɝɨ
ɧɟɨɛɯɨɞɢɦɨ ɥɢɲɶ, ɱɬɨɛɵ ɜɧɟɲɧɹɹ ɮɭɧɤɰɢɹ ɛɵɥɚ ɨɩɪɟɞɟɥɟɧɚ ɜ ɤɥɚɫɫɟ ɪɟɤɭɪɫɢɜɧɨ-ɜɵɱɢɫɥɢɦɵɯ ɮɭɧɤɰɢɣ.
ɉɪɢ ɡɚɞɚɧɧɨɣ ɜɧɟɲɧɟɣ ɢ ɩɨɞɨɛɪɚɧɧɨɣ ɜɧɭɬɪɟɧɧɟɣ ɮɭɧɤɰɢɹɯ ɜɫɟɝɞɚ ɢɦɟɟɬɫɹ ɜɨɡɦɨɠɧɨɫɬɶ ɩɨɥɭɱɢɬɶ
ɩɪɟɞɫɬɚɜɥɟɧɢɟ ɨ ɫɬɪɭɤɬɭɪɟ ɭɩɪɚɜɥɹɟɦɨɣ ɫɢɫɬɟɦɵ ɢ ɜɜɟɫɬɢ ɨɝɪɚɧɢɱɟɧɢɹ ɧɚ ɜɧɭɬɪɟɧɧɸɸ ɮɭɧɤɰɢɸ F2.
ɇɚɩɪɢɦɟɪ, ɨɝɪɚɧɢɱɟɧɢɹ, ɧɚɤɥɚɞɵɜɚɟɦɵɟ ɧɚ ɫɬɪɭɤɬɭɪɭ ɜɧɭɬɪɟɧɧɟɣ ɮɭɧɤɰɢɢ ɭɩɪɚɜɥɹɟɦɨɣ ɫɢɫɬɟɦɵ,
ɫɩɨɫɨɛɧɵ ɩɪɢɜɟɫɬɢ ɤ ɪɟɲɟɧɢɸ ɩɪɨɛɥɟɦɭ ɪɚɡɦɟɪɧɨɫɬɢ ɢɥɢ ɚɥɝɟɛɪɚɢɱɟɫɤɨɣ ɫɥɨɠɧɨɫɬɢ, ɚ ɬɚɤɠɟ ɩɪɨɛɥɟɦɵ
ɢɞɟɧɬɢɮɢɤɚɰɢɢ. ɂɦɟɧɧɨ ɦɟɬɨɞ ɤɨɧɫɬɪɭɢɪɨɜɚɧɢɹ ɫɢɫɬɟɦɵ ɭɩɪɚɜɥɟɧɢɹ ɜ ɜɢɞɟ ɨɬɨɛɪɚɠɟɧɢɹ (2) ɩɨɡɜɨɥɹɟɬ
ɫɞɟɥɚɬɶ ɩɪɨɡɪɚɱɧɨɣ ɩɪɨɰɟɞɭɪɭ ɫɨɫɬɚɜɥɟɧɢɹ ɦɚɥɨɩɚɪɚɦɟɬɪɢɱɟɫɤɢɯ ɦɚɬɟɦɚɬɢɱɟɫɤɢɯ ɦɨɞɟɥɟɣ
ɭɩɪɚɜɥɹɟɦɵɯ ɫɢɫɬɟɦ. ɉɨɷɬɨɦɭ ɩɟɪɜɨɣ ɨɫɨɛɟɧɧɨɫɬɶɸ, ɤɨɬɨɪɭɸ ɧɟɨɛɯɨɞɢɦɨ ɭɱɢɬɵɜɚɬɶ ɩɪɢ ɩɨɫɬɪɨɟɧɢɢ
ɫɢɫɬɟɦɵ ɭɩɪɚɜɥɟɧɢɹ, ɹɜɥɹɟɬɫɹ ɩɪɢɨɪɢɬɟɬ ɜɧɟɲɧɟɣ ɮɭɧɤɰɢɢ F1 ɧɚɞ ɜɧɭɬɪɟɧɧɟɣ F2.
ȼɟɞɟɧɧɵɟ ɮɭɧɤɰɢɢ F1 ɢ F2 ɢɦɟɸɬ ɚɛɫɬɪɚɤɬɧɵɣ ɯɚɪɚɤɬɟɪ, ɩɨɫɤɨɥɶɤɭ ɫɩɨɫɨɛɧɵ ɨɬɜɟɱɚɬɶ ɥɢɲɶ ɧɚ
ɞɜɚ ɞɨɫɬɚɬɨɱɧɨ ɨɛɳɢɯ ɜɨɩɪɨɫɚ, ɤɚɫɚɸɳɢɯɫɹ ɭɩɪɚɜɥɹɟɦɨɣ ɫɢɫɬɟɦɵ. ȼɨ-ɩɟɪɜɵɯ, ɡɚɱɟɦ ɫɢɫɬɟɦɚ ɫɨɡɞɚɟɬɫɹ
ɢɥɢ ɜɵɞɟɥɹɟɬɫɹ, ɚ ɜɨ-ɜɬɨɪɵɯ, ɤɚɤɢɦ ɨɛɪɚɡɨɦ ɜ ɫɢɫɬɟɦɟ ɩɨɥɭɱɚɟɬɫɹ ɠɟɥɚɟɦɵɣ ɪɟɡɭɥɶɬɚɬ. Ɉɛɴɟɞɢɧɟɧɢɟ
ɨɬɜɟɬɨɜ, ɫɮɨɪɦɭɥɢɪɨɜɚɧɧɨɟ ɫ ɩɨɦɨɳɶɸ F1 ɢ F2, ɪɚɫɤɪɵɜɚɟɬ ɨɞɧɨ ɨɛɳɟɟ ɫɜɨɣɫɬɜɨ ɫɢɫɬɟɦɵ –
ɰɟɥɟɧɚɩɪɚɜɥɟɧɧɨɟ ɞɟɣɫɬɜɢɟ, ɩɪɨɢɫɯɨɞɹɳɟɟ ɜ ɧɟɣ ɫɚɦɨɣ ɢ ɫ ɟɟ ɷɥɟɦɟɧɬɚɦɢ. ɉɪɢɱɟɦ ɞɟɣɫɬɜɢɟ ɞɨɥɠɧɨ
ɛɵɬɶ ɨɪɝɚɧɢɡɨɜɚɧɨ ɬɚɤ, ɱɬɨɛɵ ɨɧɨ ɩɨɡɜɨɥɹɥɨ ɪɟɚɥɢɡɨɜɚɬɶ ɩɨɫɬɚɜɥɟɧɧɭɸ ɰɟɥɶ, ɤɨɧɤɪɟɬɢɡɢɪɨɜɚɧɧɭɸ ɫ
ɩɨɦɨɳɶɸ F1. ɋɥɟɞɨɜɚɬɟɥɶɧɨ, ɨɬɨɛɪɚɠɟɧɢɟ ɨɛɴɟɞɢɧɟɧɢɹ ɜɧɭɬɪɟɧɧɟɣ ɮɭɧɤɰɢɢ F2 ɢ ɜɧɟɲɧɟɣ F1 ɦɨɠɧɨ
ɪɚɫɫɦɚɬɪɢɜɚɬɶ ɤɚɤ ɰɟɥɟɧɚɩɪɚɜɥɟɧɧɨɟ ɞɟɣɫɬɜɢɟ L, ɫɩɨɫɨɛɧɨɟ ɩɨɥɧɨɫɬɶɸ ɨɩɢɫɚɬɶ ɭɩɪɚɜɥɹɟɦɭɸ ɫɢɫɬɟɦɭ.
Ɂɚɦɤɧɭɬɨɫɬɶ ɜɧɟɲɧɟɣ ɢ ɜɧɭɬɪɟɧɧɟɣ ɮɭɧɤɰɢɣ, ɟɫɬɟɫɬɜɟɧɧɵɦ ɨɛɪɚɡɨɦ, ɩɨɪɨɠɞɚɟɬ ɡɚɦɤɧɭɬɨɫɬɶ
ɰɟɥɟɧɚɩɪɚɜɥɟɧɧɨɝɨ ɞɟɣɫɬɜɢɹ, ɬɚɤ ɱɬɨ
> L @  L.
ɉɨɷɬɨɦɭ ɜ ɥɸɛɵɯ ɫɢɫɬɟɦɚɯ ɭɩɪɚɜɥɟɧɢɹ, ɜ ɱɚɫɬɧɨɫɬɢ, ɛɟɡɨɩɚɫɧɨɫɬɶɸ ɦɨɪɫɤɨɝɨ ɫɭɞɨɯɨɞɫɬɜɚ ɢɥɢ
ɤɚɱɟɫɬɜɨɦ ɬɪɚɧɫɩɨɪɬɧɵɯ ɭɫɥɭɝ, ɦɨɠɧɨ ɨɯɚɪɚɤɬɟɪɢɡɨɜɚɬɶ ɬɚɤɨɟ ɞɟɣɫɬɜɢɟ ɫɥɟɞɭɸɳɢɦ ɡɚɦɤɧɭɬɵɦ ɩɨ
ɨɛɴɟɞɢɧɟɧɢɸ ɜɵɪɚɠɟɧɢɟɦ
L = SS TQ SA,
(4)
ɝɞɟ SS – ɧɟɩɭɫɬɨɟ ɫɜɹɡɚɧɧɨɟ ɢ ɡɚɦɤɧɭɬɨɟ ɦɧɨɠɟɫɬɜɨ ɷɥɟɦɟɧɬɨɜ, ɫɨɫɬɚɜɥɹɸɳɢɯ ɫɢɫɬɟɦɭ; TQ – ɨɛɳɚɹ
ɯɚɪɚɤɬɟɪɢɫɬɢɤɚ ɰɟɥɟɧɚɩɪɚɜɥɟɧɧɨɝɨ ɞɟɣɫɬɜɢɹ ɦɟɠɞɭ ɷɥɟɦɟɧɬɚɦɢ; SA – ɫɚɦɨɨɰɟɧɤɚ ɪɟɡɭɥɶɬɚɬɨɜ ɞɜɢɠɟɧɢɹ
ɫɢɫɬɟɦɵ ɜ ɨɬɧɨɲɟɧɢɢ ɰɟɥɢ ɭɩɪɚɜɥɟɧɢɹ F1.
3. ɐɟɥɟɧɚɩɪɚɜɥɟɧɧɨɟ ɞɟɣɫɬɜɢɟ ɤɚɤ ɤɥɚɫɫ ɷɤɜɢɜɚɥɟɧɬɧɨɫɬɢ
Ɉɛɵɱɧɨ ɩɨɧɹɬɢɟ ɮɭɧɤɰɢɢ ɫɬɚɜɢɬɫɹ ɜ ɡɚɜɢɫɢɦɨɫɬɶ ɨɬ ɞɟɣɫɬɜɢɹ ɢ ɩɪɨɬɢɜɨɩɨɫɬɚɜɥɹɟɬɫɹ ɩɨɧɹɬɢɸ
ɫɬɪɭɤɬɭɪɵ. ȿɫɥɢ ɠɟ ɢɫɯɨɞɢɬɶ ɢɡ ɨɩɪɟɞɟɥɟɧɢɹ ɭɩɪɚɜɥɹɟɦɨɣ ɫɢɫɬɟɦɵ ɜɢɞɚ (2) ɤɚɤ ɰɟɥɟɧɚɩɪɚɜɥɟɧɧɨɝɨ
ɞɟɣɫɬɜɢɹ (4), ɬɨ ɦɨɠɧɨ ɨɬɤɚɡɚɬɶɫɹ ɨɬ ɩɪɢɧɹɬɨɝɨ ɩɪɨɬɢɜɨɩɨɫɬɚɜɥɟɧɢɹ ɢ ɪɚɫɫɦɚɬɪɢɜɚɬɶ ɫɬɪɭɤɬɭɪɭ ɥɢɲɶ
ɤɚɤ ɩɪɟɞɫɬɚɜɥɟɧɢɟ (4), ɞɨɩɭɫɤɚɸɳɟɟ ɫɧɢɠɟɧɢɟ ɭɪɨɜɧɹ ɚɛɫɬɪɚɝɢɪɨɜɚɧɢɹ L, ɡɚɥɨɠɟɧɧɨɝɨ ɫ ɩɨɦɨɳɶɸ
ɮɭɧɤɰɢɣ F1 ɢ F2. Ⱦɥɹ ɫɧɢɠɟɧɢɹ ɭɪɨɜɧɹ ɚɛɫɬɪɚɝɢɪɨɜɚɧɢɹ ɰɟɥɟɧɚɩɪɚɜɥɟɧɧɨɝɨ ɞɟɣɫɬɜɢɹ ɫɥɟɞɭɟɬ
ɩɪɢɜɥɟɤɚɬɶ ɬɚɤɢɟ ɚɫɩɟɤɬɵ ɷɬɨɝɨ ɞɟɣɫɬɜɢɹ, ɤɚɤ ɟɝɨ ɝɟɨɦɟɬɪɢɱɟɫɤɢɟ, ɤɢɧɟɦɚɬɢɱɟɫɤɢɟ, ɦɟɯɚɧɢɱɟɫɤɢɟ,
ɮɢɡɢɱɟɫɤɢɟ, ɦɨɪɮɨɥɨɝɢɱɟɫɤɢɟ ɫɜɨɣɫɬɜɚ. ɉɪɢɱɟɦ ɜ ɫɢɫɬɟɦɚɯ ɭɩɪɚɜɥɟɧɢɹ ɜɢɞɚ (4) ɩɨɧɢɡɢɬɶ ɭɪɨɜɟɧɶ
ɚɛɫɬɪɚɝɢɪɨɜɚɧɢɹ ɦɨɠɧɨ, ɟɫɥɢ ɢɫɩɨɥɶɡɨɜɚɬɶ, ɧɚɩɪɢɦɟɪ, ɬɨɥɶɤɨ ɦɨɪɮɨɥɨɝɢɱɟɫɤɢɟ ɚɫɩɟɤɬɵ, ɤɚɤ ɛɨɥɟɟ
ɨɛɳɢɟ ɩɨ ɨɬɧɨɲɟɧɢɸ ɤ ɨɫɬɚɥɶɧɵɦ. Ɇɨɪɮɨɥɨɝɢɱɟɫɤɢɦ ɫɜɨɣɫɬɜɨɦ ɰɟɥɟɧɚɩɪɚɜɥɟɧɧɨɝɨ ɞɟɣɫɬɜɢɹ L ɜ (2)
ɛɭɞɟɦ ɧɚɡɵɜɚɬɶ ɧɟɤɨɬɨɪɨɟ ɦɧɨɠɟɫɬɜɨ ɮɢɡɢɱɟɫɤɢɯ ɫɜɨɣɫɬɜ, ɤɨɬɨɪɵɟ ɫɩɨɫɨɛɧɵ ɩɨɪɨɠɞɚɬɶ
ɮɭɧɤɰɢɨɧɚɥɶɧɵɟ ɨɬɧɨɲɟɧɢɹ ɦɟɠɞɭ ɨɬɞɟɥɶɧɵɦɢ ɷɥɟɦɟɧɬɚɦɢ ɫɢɫɬɟɦɵ. ɂɦɟɧɧɨ ɬɚɤɚɹ ɜɡɚɢɦɨɫɜɹɡɶ ɦɟɠɞɭ
ɮɭɧɤɰɢɟɣ ɢ ɫɬɪɭɤɬɭɪɨɣ ɩɨɡɜɨɥɹɟɬ ɨɫɭɳɟɫɬɜɢɬɶ ɨɩɟɪɚɰɢɸ ɞɟɤɨɦɩɨɡɢɰɢɢ ɰɟɥɟɧɚɩɪɚɜɥɟɧɧɨɝɨ ɞɟɣɫɬɜɢɹ L
ɜ ɫɢɫɬɟɦɟ (2) ɩɨ ɤɨɧɟɱɧɨɦɟɪɧɨɦɭ ɛɚɡɢɫɭ ɦɨɪɮɨɥɨɝɢɱɟɫɤɢɯ ɫɜɨɣɫɬɜ.
Ɉɱɟɜɢɞɧɨ, ɱɬɨ ɦɨɪɮɨɥɨɝɢɱɟɫɤɢɦ ɤɥɚɫɫɨɦ ɮɭɧɤɰɢɣ ɰɟɥɟɧɚɩɪɚɜɥɟɧɧɨɝɨ ɞɟɣɫɬɜɢɹ L ɜ ɫɢɫɬɟɦɟ
ɫɥɟɞɭɟɬ ɫɱɢɬɚɬɶ ɢɯ ɦɧɨɠɟɫɬɜɨ, ɤɨɬɨɪɵɟ ɫɜɹɡɚɧɨ, ɩɨ ɤɪɚɣɧɟɣ ɦɟɪɟ, ɨɞɧɢɦ ɨɛɳɢɦ ɦɨɪɮɨɥɨɝɢɱɟɫɤɢɦ
ɫɜɨɣɫɬɜɨɦ. Ɍɨɝɞɚ L ɦɨɠɧɨ ɩɪɟɞɫɬɚɜɢɬɶ ɤɚɤ ɤɥɚɫɫ ɫ ɯɚɪɚɤɬɟɪɢɫɬɢɤɨɣ, ɨɩɪɟɞɟɥɟɧɧɨɣ ɧɚ ɭɪɨɜɧɟ
ɫɨɨɬɧɨɲɟɧɢɹ ɷɤɜɢɜɚɥɟɧɬɧɨɫɬɢ ɜɢɞɚ:
A  L u L,
9
Ɇɟɧɶɲɢɤɨɜ ȼ.ɂ. ɢ ɞɪ. ɋɢɫɬɟɦɚ ɭɩɪɚɜɥɟɧɢɹ ɫ ɩɨɡɢɰɢɢ ɰɟɥɟɧɚɩɪɚɜɥɟɧɧɨɝɨ ɞɟɣɫɬɜɢɹ
ɚ ɭɩɨɪɹɞɨɱɟɧɧɚɹ ɩɚɪɚ < = < L, A > ɨɛɪɚɡɭɟɬ ɫɢɫɬɟɦɭ ɫ ɨɬɧɨɲɟɧɢɟɦ, ɨɩɢɫɵɜɚɸɳɭɸ ɷɬɨɬ ɤɥɚɫɫ, ɞɥɹ
ɤɨɬɨɪɨɝɨ A ɹɜɥɹɟɬɫɹ ɩɨɤɚɡɚɬɟɥɟɦ ɧɟɪɚɡɥɢɱɢɦɨɫɬɢ ɰɟɥɟɧɚɩɪɚɜɥɟɧɧɵɯ ɞɟɣɫɬɜɢɣ. ɋɥɟɞɨɜɚɬɟɥɶɧɨ,
ɩɨɧɹɬɢɟ ɭɩɪɚɜɥɹɟɦɨɣ ɫɢɫɬɟɦɵ (2) ɤɚɤ ɨɛɴɟɞɢɧɟɧɢɟ ɜɧɟɲɧɟɣ, ɚ ɬɚɤɠɟ ɜɧɭɬɪɟɧɧɟɣ ɮɭɧɤɰɢɢ ɜ ɫɢɥɭ
ɡɚɦɤɧɭɬɨɫɬɢ (4) ɦɨɠɧɨ ɫɨɩɨɫɬɚɜɢɬɶ ɢ ɩɪɢɪɚɜɧɹɬɶ ɤ ɩɨɧɹɬɢɸ ɨɩɟɪɚɬɨɪɚ, ɤɨɬɨɪɵɣ ɨɩɪɟɞɟɥɟɧ ɜ ɤɥɚɫɫɟ
< = < L, A > ɢ ɨɛɪɚɡɭɟɬ ɜɧɭɬɪɟɧɧɸɸ ɮɭɧɤɰɢɸ ɫɢɫɬɟɦɵ. ɉɪɢ ɷɬɨɦ ɜɧɭɬɪɟɧɧɟɟ ɰɟɥɟɧɚɩɪɚɜɥɟɧɧɨɟ
ɞɟɣɫɬɜɢɟ (2) ɧɟɨɛɯɨɞɢɦɨ ɪɚɫɫɦɚɬɪɢɜɚɬɶ ɬɨɥɶɤɨ ɫ ɦɨɪɮɨɥɨɝɢɱɟɫɤɢɯ ɩɨɡɢɰɢɣ, ɬɚɤ ɤɚɤ ɢɡɦɟɧɟɧɢɹ,
ɤɚɫɚɸɳɢɟɫɹ ɯɚɪɚɤɬɟɪɚ ɨɩɟɪɚɬɨɪɚ, ɧɟ ɜɥɢɹɸɬ ɧɚ ɫɭɳɧɨɫɬɶ ɷɬɨɝɨ ɞɟɣɫɬɜɢɹ.
ȼ ɡɚɜɢɫɢɦɨɫɬɢ ɨɬ ɜɧɟɲɧɟɣ ɮɭɧɤɰɢɢ, ɮɨɪɦɢɪɭɸɳɟɣ ɫɢɫɬɟɦɭ, ɦɨɪɮɨɥɨɝɢɹ ɤɥɚɫɫɚ < = < L, A >
ɦɨɠɟɬ ɛɵɬɶ ɞɟɤɨɦɩɨɡɢɪɨɜɚɧɚ ɤɚɤ ɜ ɭɡɤɨɦ, ɬɚɤ ɢ ɜ ɲɢɪɨɤɨɦ ɫɦɵɫɥɟ. ɍɪɨɜɟɧɶ ɞɟɤɨɦɩɨɡɢɰɢɢ
ɨɩɟɪɚɬɨɪɚ ɢɡ < = < L, A > ɫɩɨɫɨɛɟɧ ɨɛɴɹɫɧɢɬɶ ɬɚɤɨɟ ɜɚɠɧɨɟ ɩɨɧɹɬɢɟ, ɤɚɤ ɜɧɭɬɪɟɧɧɢɣ ɦɚɫɲɬɚɛ
ɫɢɫɬɟɦɵ. Ɉɩɪɟɞɟɥɟɧɢɟ ɨɩɟɪɚɬɨɪɚ ɫɢɫɬɟɦɵ (2) ɫ ɩɨɦɨɳɶɸ ɜɧɟɲɧɟɣ ɢ ɜɧɭɬɪɟɧɧɟɣ ɮɭɧɤɰɢɢ,
ɨɝɪɚɧɢɱɟɧɢɟ (4) ɢ ɮɢɤɫɚɰɢɹ ɟɝɨ ɜ ɤɥɚɫɫɟ < = < L, A > ɩɨɡɜɨɥɹɸɬ ɫɱɢɬɚɬɶ, ɱɬɨ ɜ ɞɚɧɧɨɦ ɫɥɭɱɚɢ
ɰɟɥɟɧɚɩɪɚɜɥɟɧɧɨɟ ɞɟɣɫɬɜɢɟ ɩɨɥɧɨɫɬɶɸ ɨɩɪɟɞɟɥɟɧɨ, ɚ, ɫɥɟɞɨɜɚɬɟɥɶɧɨ, ɦɨɪɮɨɥɨɝɢɱɟɫɤɢɟ ɫɜɨɣɫɬɜɚ,
ɮɨɪɦɢɪɭɸɳɢɟ ɬɚɤɨɣ ɤɥɚɫɫ ɷɤɜɢɜɚɥɟɧɬɧɨɫɬɢ, ɩɨɞɞɚɸɬɫɹ ɨɛɹɡɚɬɟɥɶɧɨɣ ɞɟɤɨɦɩɨɡɢɰɢɢ. ɉɪɨɰɟɫɫ ɠɟ
ɫɚɦɨɣ ɞɟɤɨɦɩɨɡɢɰɢɢ ɧɟɢɡɛɟɠɧɨ ɩɪɨɜɟɞɟɬ ɤ ɩɪɟɞɫɬɚɜɥɟɧɢɸ L ɢɡ < = < L, A > ɤɚɤ ɩɨɥɧɨɝɨ ɤɥɚɫɫɚ
ɧɟɩɟɪɟɫɟɤɚɸɳɢɯɫɹ ɦɨɪɮɨɥɨɝɢɱɟɫɤɢɯ ɫɜɨɣɫɬɜ ɰɟɥɟɧɚɩɪɚɜɥɟɧɧɨɝɨ ɞɟɣɫɬɜɢɹ. Ɂɞɟɫɶ ɫɥɟɞɭɟɬ ɡɚɦɟɬɢɬɶ,
ɱɬɨ ɜɫɹɤɨɟ ɦɨɪɮɨɥɨɝɢɱɟɫɤɨɟ ɞɟɣɫɬɜɢɟ ɫɜɨɞɢɬɫɹ ɤ ɬɚɤɨɦɭ ɠɟ ɦɧɨɠɟɫɬɜɭ ɮɢɡɢɱɟɫɤɢɯ ɞɟɣɫɬɜɢɣ.
ɉɨɦɢɦɨ ɷɬɨɝɨ, ɫɩɨɫɨɛ ɡɚɞɚɧɢɹ ɨɩɟɪɚɬɨɪɚ ɤɥɚɫɫɨɦ < = < L, A >, ɩɨ ɫɜɨɟɣ ɫɭɬɢ, ɪɚɜɧɨɰɟɧɟɧ
ɨɩɪɟɞɟɥɟɧɢɸ ɮɢɡɢɱɟɫɤɨɝɨ ɨɛɴɟɤɬɚ. Ɋɚɜɧɨɰɟɧɧɨɫɬɶ ɦɟɠɞɭ ɡɚɞɚɧɢɟɦ ɨɩɟɪɚɬɨɪɚ ɰɟɥɟɧɚɩɪɚɜɥɟɧɧɨɝɨ
ɞɟɣɫɬɜɢɹ ɢ ɜɵɞɟɥɟɧɢɟɦ ɨɛɴɟɤɬɚ, ɜ ɫɜɨɸ ɨɱɟɪɟɞɶ, ɩɨɡɜɨɥɹɟɬ ɧɚɞɟɹɬɶɫɹ ɧɚ ɬɨ, ɱɬɨ ɩɪɟɞɫɬɚɜɥɟɧɢɟ ɨ
ɜɧɭɬɪɟɧɧɢɯ ɫɜɨɣɫɬɜɚɯ ɫɢɫɬɟɦɵ, ɨɩɪɟɞɟɥɟɧɧɨɟ ɫ ɩɨɦɨɳɶɸ ɤɥɚɫɫɚ < = < L, A >, ɫɩɨɫɨɛɧɨ ɩɪɢɦɢɪɢɬɶ
ɨɬɦɟɱɟɧɧɭɸ ɜɵɲɟ ɩɪɨɬɢɜɨɪɟɱɢɜɨɫɬɶ ɜ ɩɨɞɯɨɞɚɯ ɤ ɤɨɧɫɬɪɭɢɪɨɜɚɧɢɸ ɩɨɧɹɬɢɹ ɫɢɫɬɟɦɵ ɞɥɹ
ɚɛɫɬɪɚɤɬɧɨɣ ɢ ɤɨɧɤɪɟɬɧɨɣ ɬɟɨɪɢɣ.
4. ɐɟɥɟɧɚɩɪɚɜɥɟɧɧɨɟ ɞɟɣɫɬɜɢɟ ɤɚɤ ɫɬɪɭɤɬɭɪɧɵɣ ɤɥɚɫɫ
ȼɜɟɞɟɧɧɵɣ ɤɥɚɫɫ ɷɤɜɢɜɚɥɟɧɬɧɨɫɬɢ ɰɟɥɟɧɚɩɪɚɜɥɟɧɧɨɝɨ ɞɟɣɫɬɜɢɹ ɢ ɟɝɨ ɞɟɤɨɦɩɨɡɢɰɢɹ ɩɨ
ɦɨɪɮɨɥɨɝɢɱɟɫɤɢɦ ɫɜɨɣɫɬɜɚɦ ɞɚɸɬ ɩɪɚɜɨ ɧɚ ɢɫɫɥɟɞɨɜɚɧɢɟ ɬɨɧɤɨɣ ɫɬɪɭɤɬɭɪɵ ɨɩɟɪɚɬɨɪɚ, ɧɨ ɥɢɲɶ ɜ
ɪɚɦɤɚɯ ɟɝɨ ɥɨɤɚɥɢɡɚɰɢɢ. Ɉɞɧɚɤɨ ɬɪɟɛɨɜɚɧɢɟ ɡɚɦɤɧɭɬɨɫɬɢ ɨɩɟɪɚɬɨɪɚ ɜɵɩɨɥɧɢɦɨ ɥɢɲɶ ɩɪɢ
ɤɨɧɫɬɚɬɚɰɢɢ ɫɜɟɞɟɧɢɣ ɨ ɫɚɦɨɦ ɡɚɦɵɤɚɧɢɢ. ɂɦɟɧɧɨ ɡɚɦɵɤɚɧɢɟ ɮɢɤɫɢɪɭɟɬ ɜɧɟɲɧɢɟ ɭɫɥɨɜɢɹ, ɩɨ
ɨɬɧɨɲɟɧɢɸ ɤ ɤɨɬɨɪɵɦ ɩɪɨɬɟɤɚɟɬ ɫɚɦɨ ɞɟɣɫɬɜɢɟ, ɨɛɟɫɩɟɱɢɜɚɹ ɤɚɤ ɪɟɚɥɢɡɚɰɢɸ ɰɟɥɢ F1, ɬɚɤ ɢ
ɧɟɨɛɯɨɞɢɦɭɸ ɜɧɭɬɪɟɧɧɸɸ ɩɪɚɤɬɢɱɟɫɤɭɸ ɞɟɹɬɟɥɶɧɨɫɬɶ. ɉɨɷɬɨɦɭ ɩɪɢ ɫɢɧɬɟɡɟ ɫɢɫɬɟɦɵ ɭɩɪɚɜɥɟɧɢɹ
ɩɨɹɜɥɹɟɬɫɹ ɧɟɨɛɯɨɞɢɦɨɫɬɶ ɜ ɪɚɫɲɢɪɟɧɢɢ ɦɨɪɮɨɥɨɝɢɱɟɫɤɨɝɨ ɤɥɚɫɫɚ ɰɟɥɟɧɚɩɪɚɜɥɟɧɧɵɯ ɞɟɣɫɬɜɢɣ ɜɢɞɚ
< L, A > ɡɚ ɫɱɟɬ ɞɨɩɨɥɧɢɬɟɥɶɧɨɝɨ ɜɤɥɸɱɟɧɢɹ ɜ ɧɟɝɨ ɝɟɨɦɟɬɪɢɱɟɫɤɢɯ, ɤɢɧɟɦɚɬɢɱɟɫɤɢɯ, ɦɟɯɚɧɢɱɟɫɤɢɯ,
ɮɢɡɢɱɟɫɤɢɯ ɢ ɦɨɪɮɨɥɨɝɢɱɟɫɤɢɯ ɚɫɩɟɤɬɨɜ ɡɚɦɵɤɚɧɢɹ ɬɚɤɨɝɨ ɞɟɣɫɬɜɢɹ. Ɋɚɫɲɢɪɟɧɧɵɣ ɡɚ ɫɱɟɬ
ɜɤɥɸɱɟɧɢɹ ɫɜɨɣɫɬɜ ɡɚɦɵɤɚɧɢɹ ɦɨɪɮɨɥɨɝɢɱɟɫɤɢɣ ɤɥɚɫɫ ɨɩɟɪɚɬɨɪɚ ɭɠɟ ɨɩɪɟɞɟɥɹɟɬ ɧɟ ɬɨɥɶɤɨ ɬɨɧɤɭɸ
ɫɬɪɭɤɬɭɪɭ ɰɟɥɟɧɚɩɪɚɜɥɟɧɧɨɝɨ ɞɟɣɫɬɜɢɹ, ɹɜɥɹɸɳɟɝɨɫɹ ɷɥɟɦɟɧɬɨɦ ɷɬɨɝɨ ɤɥɚɫɫɚ, ɚ ɞɚɟɬ ɢ ɨɛɳɟɟ
ɝɥɨɛɚɥɶɧɨɟ ɩɪɟɞɫɬɚɜɥɟɧɢɟ ɨ ɞɟɹɬɟɥɶɧɨɫɬɢ ɫɢɫɬɟɦɵ ɢ ɟɟ ɞɜɢɠɟɧɢɢ ɤ ɩɨɫɬɚɜɥɟɧɧɨɣ ɰɟɥɢ ɜ ɩɪɟɞɟɥɚɯ
ɫɬɪɭɤɬɭɪɧɵɯ ɩɪɟɞɫɬɚɜɥɟɧɢɣ ɨ ɡɚɦɵɤɚɧɢɢ. Ɍɚɤɨɣ ɨɛɨɛɳɟɧɧɵɣ ɤɥɚɫɫ ɰɟɥɟɧɚɩɪɚɜɥɟɧɧɵɯ ɞɟɣɫɬɜɢɣ
ɦɨɠɧɨ ɭɠɟ ɧɚɡɜɚɬɶ ɫɬɪɭɤɬɭɪɧɵɦ ɤɥɚɫɫɨɦ ɷɤɜɢɜɚɥɟɧɬɧɨɫɬɢ ɫɢɫɬɟɦɵ. ɏɨɬɹ ɧɟɨɛɯɨɞɢɦɨ ɨɬɦɟɬɢɬɶ, ɱɬɨ,
ɜ ɬɟɯ ɫɥɭɱɚɹɯ, ɤɨɝɞɚ ɫɬɪɭɤɬɭɪɚ ɡɚɦɵɤɚɧɢɣ F1 ɢ F2 ɨɩɪɟɞɟɥɟɧɚ ɬɨɥɶɤɨ ɧɚ ɭɪɨɜɧɟ ɦɨɪɮɨɥɨɝɢɱɟɫɤɢɯ
ɫɜɨɣɫɬɜ, ɫɬɪɭɤɬɭɪɧɵɣ ɤɥɚɫɫ ɜɵɪɨɠɞɚɟɬɫɹ ɜ ɦɨɪɮɨɥɨɝɢɱɟɫɤɢɣ.
ȿɫɥɢ ɩɪɢɧɹɬɶ, ɜɨ ɜɧɢɦɚɧɢɟ, ɱɬɨ ɦɧɨɠɟɫɬɜɨ ɞɟɣɫɬɜɢɣ >L0@  L0, ɜɤɥɸɱɚɸɳɟɟ ɜ ɫɟɛɹ ɢ
ɦɨɪɮɨɥɨɝɢɱɟɫɤɢɟ ɫɜɨɣɫɬɜɚ ɡɚɦɵɤɚɧɢɹ, ɨɛɥɚɞɚɟɬ ɨɬɧɨɲɟɧɢɟɦ ɷɤɜɢɜɚɥɟɧɬɧɨɫɬɢ R = L0 u L0, ɬɨ
ɭɩɨɪɹɞɨɱɟɧɧɚɹ ɩɚɪɚ < = < L0, R > ɛɭɞɟɬ ɯɚɪɚɤɬɟɪɢɡɨɜɚɬɶ ɨɛɳɟɟ ɩɪɨɫɬɪɚɧɫɬɜɨ ɩɪɢɛɥɢɠɟɧɢɣ ɞɥɹ L0. ɉɨ
ɫɭɬɢ, ɭɩɨɪɹɞɨɱɟɧɧɚɹ ɩɚɪɚ < = < L, R > ɩɪɟɞɫɬɚɜɥɹɟɬ ɫɨɛɨɣ ɫɢɫɬɟɦɭ ɫ ɨɬɧɨɲɟɧɢɟɦ, ɤɨɬɨɪɚɹ ɬɚɤ ɠɟ, ɤɚɤ
ɢ < L0, Ⱥ >, ɹɜɥɹɟɬɫɹ ɯɚɪɚɤɬɟɪɢɫɬɢɤɨɣ ɩɪɨɫɬɪɚɧɫɬɜɚ ɩɪɢɛɥɢɠɟɧɢɣ ɢ ɞɨɩɭɫɤɚɟɬ ɞɟɤɨɦɩɨɡɢɰɢɸ ɩɨ
ɤɥɚɫɫɚɦ ɷɤɜɢɜɚɥɟɧɬɧɨɫɬɢ, ɨɪɝɚɧɢɡɨɜɚɧɧɵɦ, ɧɚɩɪɢɦɟɪ, ɧɚ ɨɫɧɨɜɟ ɦɨɪɮɨɥɨɝɢɱɟɫɤɢɯ ɢɥɢ ɛɨɥɟɟ ɬɨɧɤɢɯ
ɮɢɡɢɱɟɫɤɢɯ ɫɜɨɣɫɬɜ ɰɟɥɟɧɚɩɪɚɜɥɟɧɧɨɝɨ ɞɟɣɫɬɜɢɹ. Ɍɚɤɢɟ ɤɥɚɫɫɵ ɦɨɠɧɨ ɪɚɫɫɦɚɬɪɢɜɚɬɶ, ɤɚɤ
ɷɥɟɦɟɧɬɚɪɧɵɟ ɦɧɨɠɟɫɬɜɚ ɜ <, ɚ ɜɫɹɤɨɟ ɢɯ ɨɛɴɟɞɢɧɟɧɢɟ ɜɧɨɜɶ ɞɨɥɠɧɨ ɩɪɢɜɟɫɬɢ ɤ ɨɛɪɚɡɨɜɚɧɢɸ
ɫɨɫɬɚɜɧɨɝɨ ɦɧɨɠɟɫɬɜɚ ɜ <, ɩɪɢɧɹɬɨɝɨ ɡɚ ɨɛɳɟɟ ɩɪɨɫɬɪɚɧɫɬɜɨ ɩɪɢɛɥɢɠɟɧɢɣ ɞɥɹ L0, ɩɪɢɱɟɦ ɜɫɟɝɞɚ
ɜɵɩɨɥɧɹɟɬɫɹ ɭɫɥɨɜɢɟ:
L  L0.
10
ȼɟɫɬɧɢɤ ɆȽɌɍ, ɬɨɦ 3, ʋ1, 2000 ɝ.
ɫɬɪ.7-12
Ɂɚɞɚɱɚ ɞɟɤɨɦɩɨɡɢɰɢɢ ɤɥɚɫɫɚ < = < L0, R >, ɬɚɤ ɠɟ ɤɚɤ ɢ ɟɝɨ ɩɨɞɤɥɚɫɫɚ < = < L, A >, ɪɟɲɚɟɦɚ
ɥɢɲɶ ɜ ɬɨɦ ɫɥɭɱɚɟ, ɤɨɝɞɚ ɞɟɣɫɬɜɢɟ, ɡɚɦɤɧɭɬɨɟ, ɧɚɩɪɢɦɟɪ, ɩɨ ɨɛɴɟɞɢɧɟɧɢɸ (4), ɞɨɩɭɫɤɚɟɬ ɜɵɞɟɥɟɧɢɟ
ɦɨɪɮɨɥɨɝɢɱɟɫɤɢɯ ɫɜɨɣɫɬɜ, ɹɜɥɹɸɳɢɯɫɹ ɛɨɥɟɟ ɬɨɧɤɢɦɢ ɩɨ ɨɬɧɨɲɟɧɢɸ ɤ ɫɨɫɬɚɜɥɹɸɳɢɦ (4). ȼ ɫɢɥɭ
ɨɩɪɟɞɟɥɟɧɢɹ ɦɨɪɮɨɥɨɝɢɱɟɫɤɢɯ ɫɜɨɣɫɬɜ ɨɛɴɟɤɬɚ ɢɥɢ ɹɜɥɟɧɢɹ, ɢɯ ɦɨɠɧɨ ɢɞɟɧɬɢɮɢɰɢɪɨɜɚɬɶ ɥɢɲɶ
ɬɨɝɞɚ, ɤɨɝɞɚ ɡɚ ɧɢɦɢ ɫɬɨɹɬ ɮɢɡɢɱɟɫɤɢɟ ɩɪɢɡɧɚɤɢ, ɨɛɥɚɞɚɸɳɢɟ ɧɢɠɧɟɣ ɢ ɜɟɪɯɧɟɣ ɝɪɚɧɹɦɢ.
Ⱦɟɬɚɥɢɡɚɰɢɹ ɦɨɪɮɨɥɨɝɢɱɟɫɤɢɯ ɫɜɨɣɫɬɜ ɫɬɪɭɤɬɭɪɧɨɝɨ ɤɥɚɫɫɚ < = < L0, R > ɢ ɢɯ ɚɤɫɢɨɦɚɬɢɡɚɰɢɹ
ɩɨɪɨɠɞɚɸɬ ɬɚɤɨɟ ɢɡɜɟɫɬɧɨɟ ɢɡ ɬɟɨɪɢɢ ɫɢɫɬɟɦ ɩɨɧɹɬɢɟ, ɤɚɤ ɦɨɞɟɥɶ.
ɋɜɨɣɫɬɜɚ ɫɬɪɭɤɬɭɪɧɨɝɨ ɤɥɚɫɫɚ < L0, R > ɩɪɟɨɛɪɚɡɭɸɬɫɹ ɜ ɬɟɨɪɢɸ ɦɨɞɟɥɢ ɭɩɪɚɜɥɹɟɦɨɣ
ɫɢɫɬɟɦɵ (ɫɢɫɬɟɦɭ ɚɤɫɢɨɦ) ɥɢɲɶ ɜ ɫɥɭɱɚɟ ɧɟɩɪɨɬɢɜɨɪɟɱɢɜɨɫɬɢ ɫɚɦɢɯ ɚɤɫɢɨɦ. ɇɟɩɪɨɬɢɜɨɪɟɱɢɜɨɫɬɶ
ɬɟɨɪɢɢ ɫɨɫɬɨɢɬ ɜ ɬɨɦ, ɱɬɨ ɢɡ ɨɛɳɧɨɫɬɢ ɚɤɫɢɨɦ ɧɟɥɶɡɹ ɩɨɥɭɱɢɬɶ ɤɚɤɨɝɨ-ɥɢɛɨ ɩɪɨɬɢɜɨɪɟɱɢɹ. Ɉɛɵɱɧɨ
ɞɥɹ ɞɨɫɬɚɬɨɱɧɨɝɨ ɲɢɪɨɤɨɝɨ ɤɥɚɫɫɚ ɮɨɪɦɚɥɶɧɵɯ ɬɟɨɪɢɣ ɭɫɥɨɜɢɟ ɧɟɩɪɨɬɢɜɨɪɟɱɢɜɨɫɬɢ ɝɪɭɩɩɵ ɚɤɫɢɨɦ
G ɪɚɜɧɨɫɢɥɶɧɨ ɫɭɳɟɫɬɜɨɜɚɧɢɸ ɜ ɢɯ ɨɛɳɧɨɫɬɢ ɯɨɬɹ ɛɵ ɨɞɧɨɝɨ ɧɟɞɨɤɚɡɭɟɦɨɝɨ ɩɪɟɞɥɨɠɟɧɢɹ. ɉɨɷɬɨɦɭ
ɫɜɨɣɫɬɜɨ ɧɟɩɪɨɬɢɜɨɪɟɱɢɜɨɫɬɢ ɜ ɚɤɫɢɨɦɚɬɢɱɟɫɤɨɣ ɝɪɭɩɩɟ G ɦɨɠɧɨ ɪɚɫɫɦɚɬɪɢɜɚɬɶ ɤɚɤ ɧɟɨɛɯɨɞɢɦɨɟ
ɞɥɹ ɫɭɳɟɫɬɜɨɜɚɧɢɹ ɫɚɦɢɯ ɦɨɞɟɥɟɣ. Ɉɞɧɚɤɨ ɬɚɤɚɹ ɧɟɨɛɯɨɞɢɦɨɫɬɶ ɫɩɨɫɨɛɧɚ ɞɚɬɶ ɨɩɢɫɚɧɢɸ ɞɟɣɫɬɜɢɹ
ɤɨɧɤɪɟɬɧɨɫɬɶ ɢ ɝɚɪɚɧɬɢɪɨɜɚɬɶ ɟɝɨ ɬɨɥɶɤɨ ɧɚ ɭɪɨɜɧɟ ɨɛɳɟɝɨ ɯɚɪɚɤɬɟɪɚ ɩɪɨɰɟɫɫɚ, ɢɞɭɳɟɝɨ ɜ ɦɨɞɟɥɹɯ
ɤɥɚɫɫɚ < = < L0, R >, ɢ ɧɟ ɛɨɥɟɟ ɬɨɝɨ. ɇɟɩɪɨɬɢɜɨɪɟɱɢɜɨɫɬɶ ɚɤɫɢɨɦɚɬɢɱɟɫɤɨɣ ɝɪɭɩɩɵ G ɧɟɫɩɨɫɨɛɧɚ ɜ
ɩɨɥɧɨɣ ɦɟɪɟ ɨɛɟɫɩɟɱɢɬɶ ɭɫɥɨɜɢɟ ɞɨɫɬɚɬɨɱɧɨɫɬɢ, ɤɨɬɨɪɨɟ ɞɟɣɫɬɜɢɬɟɥɶɧɨ ɝɚɪɚɧɬɢɪɭɟɬ, ɱɬɨ ɨɩɢɫɚɧɧɨɟ
ɞɟɣɫɬɜɢɟ ɛɭɞɟɬ ɢɦɟɟɬ ɦɟɫɬɨ. Ⱦɨɫɬɚɬɨɱɧɨɫɬɶ ɦɨɠɟɬ ɜɨɡɧɢɤɧɭɬɶ ɧɚ ɧɟɩɪɨɬɢɜɨɪɟɱɢɜɨɣ
ɚɤɫɢɨɦɚɬɢɱɟɫɤɨɣ ɝɪɭɩɩɟ ɤɥɚɫɫɚ ɷɤɜɢɜɚɥɟɧɬɧɨɫɬɢ ɥɢɲɶ ɜ ɬɟɯ ɫɥɭɱɚɹɯ, ɤɨɝɞɚ ɩɪɨɫɬɪɚɧɫɬɜɨ
ɩɪɢɛɥɢɠɟɧɢɣ < = < L0, R > ɡɚɦɤɧɭɬɨ.
Ɉɛɳɧɨɫɬɶ ɜɧɟɲɧɟɣ ɮɭɧɤɰɢɢ F1 ɞɥɹ ɜɫɟɯ ɷɥɟɦɟɧɬɨɜ ɢɡ < = < L0, R > ɢ ɥɨɤɚɥɢɡɚɰɢɹ
ɰɟɥɟɧɚɩɪɚɜɥɟɧɧɨɝɨ ɞɟɣɫɬɜɢɹ ɩɨɡɜɨɥɹɸɬ ɩɪɢɧɹɬɶ, ɱɬɨ ɧɟɩɪɨɬɢɜɨɪɟɱɢɜɨɫɬɶ ɚɤɫɢɨɦɚɬɢɱɟɫɤɨɣ ɝɪɭɩɩɵ
ɦɨɪɮɨɥɨɝɢɱɟɫɤɢɯ ɫɜɨɣɫɬɜ ɷɬɨɝɨ ɞɟɣɫɬɜɢɹ ɫɩɨɫɨɛɧɵ ɩɨɪɨɞɢɬɶ ɜ G ɧɟ ɬɨɥɶɤɨ ɧɟɨɛɯɨɞɢɦɨɫɬɶ, ɧɨ ɢ
ɞɨɫɬɚɬɨɱɧɨɫɬɶ. Ɍɨɝɞɚ ɭɫɥɨɜɢɹ ɧɟɨɛɯɨɞɢɦɨɫɬɢ ɢ ɞɨɫɬɚɬɨɱɧɨɫɬɢ ɩɪɟɜɪɚɳɚɸɬ ɝɪɭɩɩɭ G ɜ ɬɟɨɪɢɸ, ɬ. ɟ.
< L0, R > o T,
(5)
ɝɞɟ T – ɬɟɨɪɢɹ ɦɨɞɟɥɢ ɫɢɫɬɟɦ ɭɩɪɚɜɥɟɧɢɹ, ɨɩɪɟɞɟɥɹɸɳɚɹ ɩɪɨɫɬɪɚɧɫɬɜɨ ɩɪɢɛɥɢɠɟɧɢɣ <.
ɇɟ ɦɟɧɟɟ ɜɚɠɧɵɦ ɫɜɨɣɫɬɜɨɦ ɞɥɹ ɩɪɚɤɬɢɱɟɫɤɨɝɨ ɢɫɩɨɥɶɡɨɜɚɧɢɹ ɬɟɨɪɢɢ ɦɨɞɟɥɢ ɭɩɪɚɜɥɹɟɦɨɣ
ɫɢɫɬɟɦɵ ɹɜɥɹɟɬɫɹ ɟɟ ɩɨɥɧɨɬɚ. ɉɨɥɧɨɬɚ ɚɤɫɢɨɦɚɬɢɡɢɪɨɜɚɧɧɵɯ ɦɨɪɮɨɥɨɝɢɱɟɫɤɢɯ ɫɜɨɣɫɬɜ ɝɪɭɩɩɵ G
ɩɨɡɜɨɥɹɟɬ ɧɚɞɟɹɬɶɫɹ ɧɚ ɬɨ, ɱɬɨ ɜɫɟ ɪɚɫɫɦɚɬɪɢɜɚɟɦɵɟ ɩɪɨɰɟɫɫɵ ɮɭɧɤɰɢɨɧɢɪɨɜɚɧɢɹ ɫɢɫɬɟɦɵ, ɜɵɜɨɞɢɦɵɟ
ɢɡ ɬɟɨɪɢɢ T ɞɥɹ ɩɪɨɫɬɪɚɧɫɬɜɚ ɩɪɢɛɥɢɠɟɧɢɣ <, ɛɭɞɭɬ ɜɟɪɧɵ ɜ ɪɚɦɤɚɯ ɷɬɨɣ ɬɟɨɪɢɢ. Ɍɟɨɪɢɸ ɩɪɨɫɬɪɚɧɫɬɜɚ
ɩɪɢɛɥɢɠɟɧɢɣ ɦɨɠɧɨ ɫɱɢɬɚɬɶ ɩɨɥɧɨɣ, ɟɫɥɢ ɞɥɹ ɧɟɩɭɫɬɨɝɨ ɤɥɚɫɫɚ < = < L0, R > ɥɸɛɨɟ ɞɟɣɫɬɜɢɟ,
ɫɥɟɞɭɸɳɟɟ ɢɡ ɩɪɨɰɟɫɫɚ ɮɭɧɤɰɢɨɧɢɪɨɜɚɧɢɹ ɫɢɫɬɟɦɵ, ɹɜɥɹɟɬɫɹ ɢɫɬɢɧɧɵɦ ɜ ɤɚɠɞɨɣ ɨɬɞɟɥɶɧɨɣ ɦɨɞɟɥɢ.
ɉɨɥɧɨɬɚ ɤɥɚɫɫɚ < ɨɛɟɫɩɟɱɢɜɚɟɬ ɞɨɫɬɢɠɟɧɢɟ ɰɟɥɢ ɭɩɪɚɜɥɟɧɢɹ, ɢ ɜ ɷɬɨɦ ɫɦɵɫɥɟ ɨɧɚ ɞɨɥɠɧɚ ɩɪɢɡɧɚɜɚɬɶɫɹ
ɩɨɡɢɬɢɜɧɨɣ. ɍɫɥɨɜɢɹ ɧɟɨɛɯɨɞɢɦɨɫɬɢ ɢ ɞɨɫɬɚɬɨɱɧɨɫɬɢ, ɡɚɤɥɚɞɵɜɚɟɦɵɟ ɜ ɝɪɭɩɩɭ G, ɨɛɟɫɩɟɱɢɜɚɸɬ
ɩɨɡɢɬɢɜɧɭɸ ɩɨɥɧɨɬɭ ɬɟɨɪɢɢ (5).
5. Ɂɚɤɥɸɱɟɧɢɟ
Ⱥɧɚɥɢɡ ɫɬɪɭɤɬɭɪɵ < = < L0, R >, ɜɵɩɨɥɧɟɧɧɵɣ ɩɭɬɟɦ ɞɟɤɨɦɩɨɡɢɰɢɣ ɩɨ ɨɬɞɟɥɶɧɵɦ
ɦɨɪɮɨɥɨɝɢɱɟɫɤɢɦ ɫɜɨɣɫɬɜɚɦ, ɩɨɡɜɨɥɢɥ ɩɨɥɭɱɢɬɶ ɤɨɧɟɱɧɨɦɟɪɧɨɟ ɩɪɨɫɬɪɚɧɫɬɜɨ ɩɪɢɛɥɢɠɟɧɢɣ ɫ ɛɚɡɢɫɨɦ ɜ
ɜɢɞɟ ɚɤɫɢɨɦɚɬɢɡɢɪɨɜɚɧɧɨɣ ɝɪɭɩɩɵ ɫɭɠɞɟɧɢɣ, ɨɛɪɚɳɚɸɳɢɯ ɬɟɨɪɢɸ ɦɨɞɟɥɢ ɢ ɮɨɪɦɢɪɭɸɳɢɯ ɤɥɚɫɫ,
ɤɨɬɨɪɵɣ ɦɨɠɧɨ ɡɚɩɢɫɚɬɶ ɬɚɤ
Ɇ ~ mod T.
Ɍɨɝɞɚ ɩɪɢ ɪɟɲɟɧɢɢ ɩɪɚɤɬɢɱɟɫɤɢɯ ɡɚɞɚɱ ɩɪɢɜɟɞɟɧɧɨɟ ɜɵɲɟ ɫɨɨɬɧɨɲɟɧɢɟ ɞɚɟɬ ɩɪɚɜɨ
ɫɭɞɨɜɥɚɞɟɥɶɰɚɦ ɪɚɫɫɦɚɬɪɢɜɚɬɶ ɬɟɤɫɬ ɫɬɚɧɞɚɪɬɨɜ, ɤɚɤ ɨɩɢɫɚɧɢɟ ɫɜɨɣɫɬɜ ɤɥɚɫɫɚ Ɇ ɢ ɭɱɢɬɵɜɚɬɶ ɷɬɨɬ
ɮɚɤɬ ɜ ɩɪɨɰɟɫɫɟ ɫɨɫɬɚɜɥɟɧɢɹ ɪɭɤɨɜɨɞɫɬɜ ɩɨ ɭɩɪɚɜɥɟɧɢɸ ɛɟɡɨɩɚɫɧɨɫɬɶɸ ɢɥɢ ɤɚɱɟɫɬɜɨɦ ɬɪɚɧɫɩɨɪɬɧɵɯ
ɭɫɥɭɝ. ɋɨɨɬɜɟɬɫɬɜɢɟ ɬɟɤɫɬɨɜ ɫɬɚɧɞɚɪɬɨɜ ɢ ɪɭɤɨɜɨɞɫɬɜ ɨɛɟɫɩɟɱɢɜɚɟɬ ɩɨɫɥɟɞɧɢɦ ɩɨɡɢɬɢɜɧɭɸ ɩɨɥɧɨɬɭ,
ɝɚɪɚɧɬɢɪɭɸɳɭɸ, ɱɬɨ ɞɟɣɫɬɜɢɟ (2) ɜɫɟɝɞɚ ɢɦɟɟɬ ɦɟɫɬɨ ɜ ɫɢɫɬɟɦɟ ɭɩɪɚɜɥɟɧɢɹ ɢ ɧɚɩɪɚɜɥɹɟɬ ɟɟ
ɫɨɫɬɨɹɧɢɟ ɤ ɡɚɞɚɧɧɨɣ ɰɟɥɢ, ɫɮɨɪɦɭɥɢɪɨɜɚɧɧɨɣ ɜ ɩɨɥɢɬɢɤɟ ɫɭɞɨɯɨɞɧɨɣ ɤɨɦɩɚɧɢɢ.
Ʌɢɬɟɪɚɬɭɪɚ
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Ɇɟɧɶɲɢɤɨɜ ȼ.ɂ. ɢ ɞɪ. ɋɢɫɬɟɦɚ ɭɩɪɚɜɥɟɧɢɹ ɫ ɩɨɡɢɰɢɢ ɰɟɥɟɧɚɩɪɚɜɥɟɧɧɨɝɨ ɞɟɣɫɬɜɢɹ
Ɇɟɧɶɲɢɤɨɜ ȼ.ɂ., Ƚɥɭɳɟɧɤɨ ȼ.Ɇ. ɍɩɪɚɜɥɟɧɢɟ ɫɢɫɬɟɦɨɣ ɛɟɡɨɩɚɫɧɨɫɬɢ ɦɨɪɫɤɨɝɨ ɫɭɞɨɯɨɞɫɬɜɚ. ȼɟɫɬɧɢɤ
ɆȽɌɍ, ɬ.1, ʋ 1, ɫ.13-16, 1998.
ɉɚɥɟɯɨɜ ɋ.ȼ. Ɋɨɫɫɢɣɫɤɚɹ ɢɧɬɟɪɩɪɟɬɚɰɢɹ ɦɟɠɞɭɧɚɪɨɞɧɨɝɨ ɤɨɞɟɤɫɚ ɩɨ ɭɩɪɚɜɥɟɧɢɸ ɛɟɡɨɩɚɫɧɨɫɬɶɸ
(ɆɄɍȻ). Ⱦɢɫɫɟɪɬɚɰɢɹ ɜ ɮɨɪɦɟ ɧɚɭɱɧɨɝɨ ɞɨɤɥɚɞɚ ɧɚ ɫɨɢɫɤɚɧɢɟ ɫɬɟɩɟɧɢ ɞɨɤɬɨɪɚ ɬɪɚɧɫɩɨɪɬɚ, Ɇ.,
83 ɫ., 1998.
ɗɧɰɢɤɥɨɩɟɞɢɹ ɤɢɛɟɪɧɟɬɢɤɢ. Ʉɢɟɜ, Ƚɥɚɜɧɚɹ ɪɟɞɚɤɰɢɹ ɍɤɪɚɢɧɫɤɨɣ ɋɨɜɟɬɫɤɨɣ ɗɧɰɢɤɥɨɩɟɞɢɢ, 618 ɫ., 1974.
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