pril
.pdf
ɉɪɢɥɨɠɟɧɢɟ 1.
Ɉ ȼȿɊɈəɌɇɈɋɌɂ
ɋɨɡɞɚɧɢɟ ɢɧɞɭɤɬɢɜɧɨɣ ɥɨɝɢɤɢ ɫɨɩɭɬɫɬɜɨɜɚɥɨ ɢɡɭɱɟɧɢɸ ɩɪɨɫɬɟɣɲɢɯ ɮɨɪɦ ɞɜɢɠɟɧɢɹ ɦɚɬɟɪɢɢ – ɦɚɤɪɨɬɟɥ, ɡɟɦɧɵɯ ɢ ɧɟɛɟɫɧɵɯ. Ʉɚɤ ɩɪɚɜɢɥɨ, ɩɪɟɞɩɨɥɚɝɚɥɨɫɶ, ɱɬɨ ɩɪɢɱɢɧɧɵɟ ɡɚɜɢɫɢɦɨɫɬɢ ɡɞɟɫɶ – ɨɞɧɨɡɧɚɱɧɵɟ, ɚ ɫɥɭɱɚɣɧɨɫɬɶ ɦɨɠɧɨ ɢɝɧɨɪɢɪɨɜɚɬɶ. ɗɬɨ ɡɧɚɱɢɬ, ɱɬɨ ɩɪɢ ɧɟɤɨɬɨɪɵɯ ɭɫɥɨɜɢɹɯ ɞɚɧɧɨɟ ɹɜɥɟɧɢɟ ɜɫɟɝɞɚ ɩɪɨɢɫɯɨɞɢɬ.
Ɉɞɧɚɤɨ ɩɪɢ ɢɡɭɱɟɧɢɢ ɦɢɤɪɨɦɢɪɚ, ɚ ɬɚɤɠɟ ɫɨɰɢɚɥɶɧɵɯ ɡɚɤɨɧɨɦɟɪɧɨɫɬɟɣ ɢɫɫɥɟɞɨɜɚɬɟɥɢ ɫɬɨɥɤɧɭɥɢɫɶ ɫ ɹɜɥɟɧɢɹɦɢ, ɤɨɬɨɪɵɟ ɩɪɢ ɡɚɞɚɧɧɵɯ ɭɫɥɨɜɢɹɯ ɦɨɝɭɬ ɩɪɨɢɫɯɨɞɢɬɶ, ɚ ɦɨɝɭɬ ɢ ɧɟ ɩɪɨɢɫɯɨɞɢɬɶ. Ɂɞɟɫɶ ɧɟɨɛɯɨɞɢɦɨɫɬɶ ɩɪɨɛɢɜɚɟɬɫɹ ɱɟɪɟɡ ɫɥɭɱɚɣɧɨɫɬɶ, ɤɨɬɨɪɨɣ ɜ ɩɪɢɧɰɢɩɟ ɩɪɟɧɟɛɪɟɝɚɬɶ ɧɟɥɶɡɹ, ɚ ɢɡɭɱɚɟɦɵɟ ɫɜɹɡɢ ɯɚɪɚɤɬɟɪɢɡɭɸɬɫɹ ɦɧɨɠɟɫɬɜɟɧɧɨɫɬɶɸ ɩɪɢɱɢɧ ɢ ɫɥɟɞɫɬɜɢɣ. Ⱦɥɹ ɢɯ ɨɩɢɫɚɧɢɹ ɜɨɡɧɢɤɥɚ ɧɟɨɛɯɨɞɢɦɨɫɬɶ ɩɟɪɟɯɨɞɚ ɤ ɧɨɜɵɦ, ɫɬɚɬɢɫɬɢɱɟɫɤɢɦ ɦɟɬɨɞɚɦ,
ȼɵɜɨɞɵ ɫɬɚɬɢɫɬɢɤɢ ɧɟ ɨɛɥɚɞɚɸɬ ɛɟɡɭɫɥɨɜɧɨɫɬɶɸ, ɜɫɤɪɵɜɚɟɦɵɟ ɡɚɤɨɧɨɦɟɪɧɨɫɬɢ ɩɪɨɹɜɥɹɸɬɫɹ ɧɟ ɜ ɤɚɠɞɨɦ ɨɬɞɟɥɶɧɨɦ ɫɥɭɱɚɟ, ɚ ɥɢɲɶ ɜ ɦɚɫɫɟ ɨɞɧɨɬɢɩɧɵɯ ɹɜɥɟɧɢɣ. ɋɬɚɬɢɫɬɢɱɟɫɤɢɟ ɡɚɤɨɧɨɦɟɪɧɨɫɬɢ – ɷɬɨ ɡɚɤɨɧɨɦɟɪɧɨɫɬɢ ɦɚɫɫɨɜɵɯ ɹɜɥɟɧɢɣ. ȼɵɪɚɠɚɸɬɫɹ ɨɧɢ ɫ ɩɨɦɨɳɶɸ ɫɩɟɰɢɚɥɶɧɵɯ ɤɚɬɟɝɨɪɢɣ ɫɬɚɬɢɫɬɢɱɟɫɤɨɣ ɧɚɭɤɢ - ɬɚɤɢɯ ɤɚɤ ɫɪɟɞɧɢɟ, ɦɟɪɵ ɜɚɪɢɚɰɢɢ, ɩɨɤɚɡɚɬɟɥɢ ɬɟɫɧɨɬɵ ɫɜɹɡɢ ɢ ɬ.ɞ. ȼ ɦɚɫɫɨɜɵɯ ɹɜɥɟɧɢɹɯ ɜɡɚɢɦɨɫɜɹɡɢ ɩɪɨɹɜɥɹɸɬɫɹ ɜ ɜɢɞɟ ɬɟɧɞɟɧɰɢɣ, ɜ ɜɢɞɟ ɢɡɦɟɧɟɧɢɹ ɫɪɟɞɧɢɯ ɜɟɥɢɱɢɧ ɩɪɢ ɜɡɚɢɦɧɨɦ ɩɨɝɚɲɟɧɢɢ ɫɥɭɱɚɣɧɨɫɬɟɣ.
ɉɪɨɬɢɜɨɩɨɫɬɚɜɥɟɧɢɟ ɫɬɚɬɢɫɬɢɱɟɫɤɢɯ ɡɚɤɨɧɨɦɟɪɧɨɫɬɟɣ ɤɥɚɫɫɢɱɟɫɤɢɦ ɧɨɫɢɬ ɭɫɥɨɜɧɵɣ ɯɚɪɚɤɬɟɪ. Ɍɚɤ, ɫɩɪɚɜɟɞɥɢɜɨ ɡɚɦɟɬɢɥ ɇ. ȼɢɧɟɪ, ɜ ɪɚɛɨɬɚɯ ɨɫɧɨɜɚɬɟɥɹ ɤɥɚɫɫɢɱɟɫɤɨɣ ɮɢɡɢɤɢ ɂ. ɇɶɸɬɨɧɚ «ɫɨɞɟɪɠɚɥɚɫɶ ɜɚɠɧɚɹ ɫɬɚɬɢɫɬɢɱɟɫɤɚɹ ɨɝɨɜɨɪɤɚ»2, ɧɟɞɨɨɰɟɧɟɧɧɚɹ ɫɨɜɪɟɦɟɧɧɢɤɚɦɢ ɢ ɩɨɫɥɟɞɨɜɚɬɟɥɹɦɢ ɜɟɥɢɤɨɝɨ ɭɱɟɧɨɝɨ. ɇɚ ɷɬɨ ɠɟ ɨɛɫɬɨɹɬɟɥɶɫɬɜɨ ɭɤɚɡɵɜɚɥ ɟɳɟ Ɇ. Ⱦɪɨɛɢɲ, ɨɬɦɟɱɚɜɲɢɣ ɫɬɚɬɢɫɬɢɱɟɫɤɢɣ ɯɚɪɚɤɬɟɪ ɞɚɠɟ ɡɚɤɨɧɨɜ Ʉɟɩɥɟɪɚ: ɨɧɢ «...ɨɩɪɟɞɟɥɹɸɬ ɬɨɥɶɤɨ ɫɪɟɞɧɢɟ ɩɭɬɢ ɞɜɢɠɟɧɢɹ ɩɥɚɧɟɬ, ɨɬ ɤɨɬɨɪɵɯ ɩɨɫɥɟɞɧɢɟ ɩɨɫɬɨɹɧɧɨ ɭɤɥɨɧɹɸɬɫɹ ɬɨ ɜ ɨɞɧɭ, ɬɨ ɜ ɞɪɭɝɭɸ ɫɬɨɪɨɧɭ»3.
Ʌɨɝɢɱɟɫɤɨɣ ɨɫɧɨɜɨɣ ɫɬɚɬɢɫɬɢɱɟɫɤɢɯ ɦɟɬɨɞɨɜ ɢɫɫɥɟɞɨɜɚɧɢɹ ɹɜɥɹɟɬɫɹ ɜɟɪɨɹɬɧɨɫɬɶ. ɑɬɨɛɵ ɜ ɫɚɦɵɯ ɨɛɳɢɯ ɱɟɪɬɚɯ ɨɡɧɚɤɨɦɢɬɶɫɹ ɫ ɷɬɢɦ ɮɭɧɞɚɦɟɧɬɚɥɶɧɵɦ ɩɨɧɹɬɢɟɦ ɫɨɜɪɟɦɟɧɧɨɣ ɧɚɭɤɢ, ɨɛɪɚɬɢɦɫɹ ɤ ɪɚɫɫɦɨɬɪɟɧɢɸ ɫɨɛɵɬɢɣ. ɉɪɟɞɩɨɥɨɠɢɦ, ɱɬɨ ɢɦɟɟɬɫɹ ɧɟɤɨɬɨɪɵɣ, ɜɩɨɥɧɟ ɨɩɪɟɞɟɥɟɧɧɵɣ ɤɨɦɩɥɟɤɫ ɭɫɥɨɜɢɣ. ȼ ɷɬɢɯ ɭɫɥɨɜɢɹɯ ɨɫɭɳɟɫɬɜɥɹɟɬɫɹ ɫɟɪɢɹ ɢɫɩɵɬɚɧɢɣ, ɪɟɡɭɥɶɬɚɬɵ ɤɨɬɨɪɵɯ ɫɭɬɶ ɧɟɤɨɬɨɪɵɟ ɫɨɛɵɬɢɹ.
[248]
ȿɫɥɢ ɫɨɛɵɬɢɟ ɧɟɢɡɛɟɠɧɨ ɩɪɨɢɫɯɨɞɢɬ ɩɪɢ ɞɚɧɧɵɯ ɭɫɥɨɜɢɹɯ, ɨɧɨ ɧɚɡɵɜɚɟɬɫɹ ɞɨɫɬɨɜɟɪɧɵɦ. ɇɚɩɪɢɦɟɪ, ɩɪɢ ɧɨɪɦɚɥɶɧɨɦ ɞɚɜɥɟɧɢɢ – 760 ɦɦ ɪɬɭɬɧɨɝɨ ɫɬɨɥɛɚ - ɢ ɬɟɦɩɟɪɚɬɭɪɟ 100°ɋ ɯɢɦɢɱɟɫɤɢ ɱɢɫɬɚɹ ɜɨɞɚ (ɷɬɨɬ ɩɟɪɟɱɟɧɶ – ɤɨɦɩɥɟɤɫ ɭɫɥɨɜɢɣ) ɜɫɟɝɞɚ ɡɚɤɢɩɚɟɬ (ɞɨɫɬɨɜɟɪɧɨɟ ɫɨɛɵɬɢɟ).
ɋɭɳɟɫɬɜɭɸɬ ɫɨɛɵɬɢɹ, ɤɨɬɨɪɵɟ ɩɪɢ ɡɚɞɚɧɧɵɯ ɭɫɥɨɜɢɹɯ ɦɨɝɭɬ ɩɪɨɢɡɨɣɬɢ, ɚ ɦɨɝɭɬ ɢ ɧɟ ɩɪɨɢɡɨɣɬɢ. Ɍɚɤɢɟ ɫɨɛɵɬɢɹ ɦɵ ɛɭɞɟɦ ɧɚɡɵɜɚɬɶ ɫɥɭɱɚɣɧɵɦɢ.
Ʉɥɚɫɫɢɱɟɫɤɢɟ ɩɪɢɦɟɪɵ ɫɥɭɱɚɣɧɵɯ ɫɨɛɵɬɢɣ - ɜɵɩɚɞɟɧɢɟ ɝɟɪɛɚ ɩɪɢ ɛɪɨɫɚɧɢɢ ɦɨɧɟɬɵ, ɢɡɜɥɟɱɟɧɢɟ ɬɭɡɚ ɢɡ ɤɨɥɨɞɵ ɤɚɪɬ, ɜɵɩɚɞɟɧɢɟ ɲɟɫɬɟɪɤɢ ɩɪɢ ɛɪɨɫɚɧɢɢ ɤɨɫɬɢ ɢ ɬ.ɞ. Ⱦɪɭɝɢɟ ɩɪɢɦɟɪɵ – ɩɨɫɬɭɩɥɟɧɢɟ ɜɵɩɭɫɤɧɢɤɚ ɫɪɟɞɧɟɣ ɲɤɨɥɵ ɜ ɜɭɡ, ɭɞɨɜɥɟɬɜɨɪɟɧɧɨɫɬɶ ɪɚɛɨɬɧɢɤɚ, ɜɵɛɢɪɚɟɦɨɝɨ ɧɚɭɝɚɞ ɢɡ ɧɟɤɨɬɨɪɨɝɨ ɤɨɥɥɟɤɬɢɜɚ, ɫɜɨɟɣ ɫɩɟɰɢɚɥɶɧɨɫɬɶɸ ɢ ɬ.ɩ.
ȿɫɥɢ ɩɪɢ ɞɚɧɧɨɦ ɤɨɦɩɥɟɤɫɟ ɭɫɥɨɜɢɣ ɧɟɤɨɬɨɪɨɟ ɫɨɛɵɬɢɟ ɡɚɜɟɞɨɦɨ ɧɟ ɦɨɠɟɬ ɩɪɨɢɡɨɣɬɢ, ɨɧɨ ɧɚɡɵɜɚɟɬɫɹ ɧɟɜɨɡɦɨɠɧɵɦ. ɇɚɩɪɢɦɟɪ, ɡɚɬɜɟɪɞɟɜɚɧɢɟ ɜɨɞɵ (ɫɨɛɵɬɢɟ) ɩɪɢ ɪɟɚɥɢɡɚɰɢɢ ɜɵɲɟ ɭɩɨɦɹɧɭɬɵɯ ɭɫɥɨɜɢɣ. Ɋɚɡɭɦɟɟɬɫɹ, ɩɪɢ ɞɪɭɝɢɯ ɭɫɥɨɜɢɹɯ ɷɬɨ ɠɟ ɫɨɛɵɬɢɟ ɩɟɪɟɫɬɚɟɬ ɛɵɬɶ ɧɟɜɨɡɦɨɠɧɵɦ. Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, ɨɩɪɟɞɟɥɟɧɢɟ ɫɨɛɵɬɢɹ ɫɨɨɬɧɨɫɢɬɫɹ ɫ ɧɟɤɨɬɨɪɵɦ ɤɨɦɩɥɟɤɫɨɦ ɭɫɥɨɜɢɣ.
2 ȼɢɧɟɪ ɇ. Ʉɢɛɟɪɧɟɬɢɤɚ ɢ ɨɛɳɟɫɬɜɨ. Ɇ., 1958, ɫ. 24.
3 Ⱦɪɨɛɢɲ Ɇ. ɇɪɚɜɫɬɜɟɧɧɚɹ ɫɬɚɬɢɫɬɢɤɚ. ɋɉɛ., 1867, ɫ. 7.
175
ȿɫɥɢ ɞɚɧɧɵɣ ɤɨɦɩɥɟɤɫ ɭɫɥɨɜɢɣ ɪɟɚɥɢɡɭɟɬɫɹ ɦɧɨɝɨɤɪɚɬɧɨ, ɬɨ ɫɬɚɧɨɜɢɬɫɹ ɜɨɡɦɨɠɧɵɦ ɧɟ ɬɨɥɶɤɨ ɤɨɧɫɬɚɬɢɪɨɜɚɬɶ ɫɥɭɱɚɣɧɨɫɬɶ ɫɨɛɵɬɢɹ Ⱥ, ɧɨ ɢ ɤɨɥɢɱɟɫɬɜɟɧɧɨ ɨɰɟɧɢɬɶ ɜɨɡɦɨɠɧɨɫɬɶ ɟɝɨ ɩɨɹɜɥɟɧɢɹ,
Ⱦɥɢɬɟɥɶɧɵɟ ɨɩɵɬɧɵɟ ɧɚɛɥɸɞɟɧɢɹ ɧɚɞ ɩɨɹɜɥɟɧɢɟɦ ɫɥɭɱɚɣɧɨɝɨ ɫɨɛɵɬɢɹ (ɩɪɢ ɧɟɢɡɦɟɧɧɨɦ ɤɨɦɩɥɟɤɫɟ ɭɫɥɨɜɢɣ) ɩɨɤɚɡɵɜɚɸɬ, ɱɬɨ ɞɥɹ ɞɨɜɨɥɶɧɨ ɲɢɪɨɤɨɝɨ ɤɪɭɝɚ ɹɜɥɟɧɢɣ ɱɢɫɥɨ ɩɨɹɜɥɟɧɢɣ ɫɨɛɵɬɢɹ ɩɨɞɱɢɧɹɟɬɫɹ ɭɫɬɨɣɱɢɜɵɦ ɡɚɤɨɧɨɦɟɪɧɨɫɬɹɦ.
ɉɭɫɬɶ n - ɨɛɳɟɟ ɱɢɫɥɨ ɢɫɩɵɬɚɧɢɣ, ɚ m - ɱɢɫɥɨ ɬɟɯ ɢɡ ɧɢɯ, ɤɨɬɨɪɵɟ ɡɚɜɟɪɲɢɥɢɫɶ ɩɨɹɜɥɟɧɢɟɦ ɫɥɭɱɚɣɧɨɝɨ ɫɨɛɵɬɢɹ Ⱥ. ȿɫɥɢ ɩɪɨɜɨɞɢɬɶ ɛɨɥɶɲɨɟ ɱɢɫɥɨ ɧɟɡɚɜɢɫɢɦɵɯ ɢɫɩɵɬɚɧɢɣ
ɜ ɧɟɢɡɦɟɧɧɵɯ ɭɫɥɨɜɢɹɯ, ɬɨ, ɤɚɤ ɩɨɤɚɡɵɜɚɟɬ ɨɩɵɬ, ɱɚɫɬɨɫɬɶ v m ɧɟɡɧɚɱɢɬɟɥɶɧɨ ɨɬɤɥɨɧɹɟɬɫɹ n
ɨɬ ɧɟɤɨɬɨɪɨɝɨ ɱɢɫɥɚ ɪ, ɤɨɬɨɪɨɟ, ɩɨ ɨɩɪɟɞɟɥɟɧɢɸ, ɧɚɡɵɜɚɟɬɫɹ ɜɟɪɨɹɬɧɨɫɬɶɸ Ⱥ. ɑɟɦ ɛɨɥɶɲɟ ɩɪɨɜɟɞɟɧɨ ɢɫɩɵɬɚɧɢɣ, ɬɟɦ ɛɥɢɠɟ ɱɚɫɬɨɫɬɶ ɤ ɜɟɪɨɹɬɧɨɫɬɢ.
ȼɟɪɨɹɬɧɨɫɬɶ, ɬɚɤɢɦ ɨɛɪɚɡɨɦ, ɨɩɪɟɞɟɥɹɟɬɫɹ ɤɚɤ ɫɪɟɞɧɹɹ ɱɚɫɬɨɫɬɶ ɩɪɢ ɛɨɥɶɲɨɦ ɱɢɫɥɟ
ɢɫɩɵɬɚɧɢɣ ɢɥɢ ɬɨɱɧɟɟ: p lim m ɇɚɩɪɢɦɟɪ, ɩɪɢ ɛɪɨɫɚɧɢɢ ɦɨɧɟɬɵ Ȼɸɮɮɨɧ ɞɥɹ n = 4040
nof n
ɩɨɥɭɱɢɥ ɱɚɫɬɨɫɬɶ ɜɵɩɚɞɟɧɢɹ ɝɟɪɛɚ v = 0,5080, ɉɢɪɫɨɧ ɞɥɹ ɩ = 12000 v = 0,5016, ɚ ɞɥɹ n = 24000 v = 0,5005. ȼ ɤɚɱɟɫɬɜɟ v ɞɚɧɧɨɦ ɫɥɭɱɚɟ ɩɪɢɧɢɦɚɟɬɫɹ 0,5. Ʌɸɛɨɩɵɬɧɨ, ɱɬɨ ɜɩɟɪɜɵɟ ɭɫɬɨɣɱɢɜɨɫɬɶ ɱɚɫɬɨɫɬɢ ɛɵɥɚ ɨɛɧɚɪɭɠɟɧɚ ɩɪɢ ɢɡɭɱɟɧɢɢ ɞɟɦɨɝɪɚɮɢɱɟɫɤɢɯ ɹɜɥɟɧɢɣ ɧɚ ɛɨɥɶɲɨɣ ɫɬɚɬɢɫɬɢɤɟ.
ȼɩɨɫɥɟɞɫɬɜɢɢ ɛɵɥɨ ɬɚɤɠɟ ɭɫɬɚɧɨɜɥɟɧɨ, ɱɬɨ ɪɚɫɩɪɟɞɟɥɟɧɢɟ ɩɨ ɪɨɫɬɭ, ɲɢɪɢɧɟ ɝɪɭɞɧɨɣ ɤɥɟɬɤɢ, ɞɥɢɧɟ ɫɬɭɩɧɢ ɥɸɞɟɣ ɨɩɪɟɞɟɥɟɧɧɨɝɨ ɩɨɥɚ, ɜɨɡɪɚɫɬɚ ɢ ɧɚɰɢɨɧɚɥɶɧɨɫɬɢ ɩɨɞɱɢɧɹɟɬɫɹ ɭɫɬɨɣɱɢɜɨɣ ɡɚɤɨɧɨɦɟɪɧɨɫɬɢ. Ʌɸɛɨɩɵɬɧɵɣ ɩɪɢɦɟɪ ɩɪɢɜɨɞɢɬ ɜ ɫɜɨɟɣ ɤɧɢɝɟ «Ɉɩɵɬ ɮɢɥɨɫɨɮɢɢ ɬɟɨɪɢɢ ɜɟɪɨɹɬɧɨɫɬɟɣ» Ʌɚɩɥɚɫ. ɂɡɭɱɚɹ ɡɚɤɨɧɨɦɟɪɧɨɫɬɢ ɪɨɠɞɟɧɢɹ ɦɚɥɶɱɢɤɨɜ ɢ ɞɟɜɨɱɟɤ, ɨɧ ɨɛɧɚɪɭɠɢɥ, ɱɬɨ ɞɥɹ ɫɬɚɬɢɫɬɢɱɟɫɤɢɯ ɦɚɬɟɪɢɚɥɨɜ Ʌɨɧɞɨɧɚ, Ȼɟɪɥɢɧɚ, ɉɟɬɟɪɛɭɪɝɚ, Ɏɪɚɧɰɢɢ (ɜ ɰɟɥɨɦ) ɨɬɧɨɫɢɬɟɥɶɧɵɟ ɱɚɫɬɨɬɵ ɪɨɠɞɟɧɢɹ ɦɚɥɶɱɢɤɨɜ ɜ ɬɟɱɟɧɢɟ ɞɟɫɹɬɢɥɟɬɢɣ
ɤɨɥɟɛɥɸɬɫɹ ɨɤɨɥɨ: 22 | 0.512 Ⱦɥɹ ɫɚɦɨɝɨ ɉɚɪɢɠɚ, ɨɞɧɚɤɨ, ɩɨɥɭɱɚɥɚɫɶ ɧɟɫɤɨɥɶɤɨ ɦɟɧɶɲɚɹ
43
ɰɢɮɪɚ: 25 | 0.510. Ʌɚɩɥɚɫ ɡɚɢɧɬɟɪɟɫɨɜɚɥɫɹ ɷɬɢɦ ɪɚɡɥɢɱɢɟɦ (ɜ ɞɜɟ ɬɵɫɹɱɧɵɯ!) ɢ ɨɛɧɚɪɭɠɢɥ, 49
ɱɬɨ ɜ ɨɛɳɟɟ ɱɢɫɥɨ ɪɨɠɞɟɧɢɣ ɜɨ ɮɪɚɧɰɭɡɫɤɨɣ ɫɬɨɥɢɰɟ ɜɤɥɸɱɚɸɬɫɹ ɩɨɞɤɢɞɵɲɢ; ɜɵɹɫɧɢɥɨɫɶ ɬɚɤɠɟ, ɱɬɨ ɨɤɪɟɫɬɧɨɟ ɫɟɥɶɫɤɨɟ ɧɚɫɟɥɟɧɢɟ
[249]
ɩɪɟɢɦɭɳɟɫɬɜɟɧɧɨ ɩɨɞɤɢɞɵɜɚɟɬ ɞɟɜɨɱɟɤ, ɱɬɨ ɢ ɢɫɤɚɡɢɥɨ ɤɚɪɬɢɧɭ. ɂɫɤɥɸɱɢɜ ɩɨɞɤɢɞɵɲɟɣ, Ʌɚɩɥɚɫ ɢ ɞɥɹ ɉɚɪɢɠɚ ɩɨɥɭɱɢɥ 22/43. Ɉɱɟɜɢɞɧɨ, ɷɬɨ ɱɢɫɥɨ ɢ ɟɫɬɶ ɜɟɪɨɹɬɧɨɫɬɶ ɪɨɠɞɟɧɢɹ ɦɚɥɶɱɢɤɚ.
ɂɡ ɨɩɪɟɞɟɥɟɧɢɹ ɜɟɪɨɹɬɧɨɫɬɢ ɫɨɛɵɬɢɹ Ⱥ ɫɥɟɞɭɟɬ, ɱɬɨ 0 d P A d1, ɩɪɢɱɟɦ 0 ɫɨɨɬɜɟɬɫɬɜɭɟɬ ɧɟɜɨɡɦɨɠɧɨɦɭ ɫɨɛɵɬɢɸ, 1 – ɞɨɫɬɨɜɟɪɧɨɦɭ.
ɂɬɚɤ, ɫɨɝɥɚɫɧɨ ɧɚɲɟɣ ɫɯɟɦɟ, ɜ ɨɞɧɢɯ ɢ ɬɟɯ ɠɟ ɭɫɥɨɜɢɹɯ ɦɨɠɧɨ ɩɪɨɜɟɫɬɢ ɧɟɨɝɪɚɧɢɱɟɧɧɨɟ ɱɢɫɥɨ ɢɫɩɵɬɚɧɢɣ, ɜ ɤɚɠɞɨɦ ɢɡ ɤɨɬɨɪɵɯ ɫɨɛɵɬɢɟ Ⱥ ɦɨɠɟɬ ɩɨɹɜɢɬɶɫɹ, ɚ ɦɨɠɟɬ ɢ ɧɟ ɩɨɹɜɢɬɶɫɹ; ɜ ɪɟɡɭɥɶɬɚɬɟ ɛɨɥɶɲɨɝɨ ɱɢɫɥɚ ɢɫɩɵɬɚɧɢɣ ɭɫɬɚɧɚɜɥɢɜɚɟɬɫɹ, ɱɬɨ ɱɚɫɬɨɬɚ ɩɨɹɜɥɟɧɢɣ ɫɨɛɵɬɢɹ Ⱥ ɞɥɹ ɤɚɠɞɨɣ ɛɨɥɶɲɨɣ ɝɪɭɩɩɵ ɢɫɩɵɬɚɧɢɣ ɦɚɥɨ ɨɬɥɢɱɚɟɬɫɹ ɨɬ ɧɟɤɨɬɨɪɨɣ ɩɨɫɬɨɹɧɧɨɣ ɜɟɥɢɱɢɧɵ, ɡɧɚɱɟɧɢɟ ɷɬɨɣ ɩɨɫɬɨɹɧɧɨɣ, ɩɨ ɨɩɪɟɞɟɥɟɧɢɸ, ɧɚɡɵɜɚɟɬɫɹ
ɫɬɚɬɢɫɬɢɱɟɫɤɨɣ ɜɟɪɨɹɬɧɨɫɬɶɸ.
ɉɪɢ ɫɬɚɬɢɫɬɢɱɟɫɤɨɦ ɨɩɪɟɞɟɥɟɧɢɢ ɞɥɹ ɧɚɯɨɠɞɟɧɢɹ ɜɟɪɨɹɬɧɨɫɬɢ ɧɟɨɛɯɨɞɢɦɨ ɩɪɨɜɟɞɟɧɢɟ ɛɨɥɶɲɨɝɨ ɱɢɫɥɚ ɢɫɩɵɬɚɧɢɣ (Ȼɸɮɮɨɧ, ɉɢɪɫɨɧ, Ʌɚɩɥɚɫ). ȼ ɪɹɞɟ ɫɥɭɱɚɟɜ ɨɤɚɡɵɜɚɟɬɫɹ ɜɨɡɦɨɠɧɵɦ ɞɚɬɶ ɚɩɪɢɨɪɧɨɟ ɨɩɪɟɞɟɥɟɧɢɟ ɜɟɪɨɹɬɧɨɫɬɢ ɫɨɛɵɬɢɹ, ɬ.ɟ. ɛɟɡ ɮɚɤɬɢɱɟɫɤɢɯ ɢɫɩɵɬɚɧɢɣ.
Ⱦɥɹ ɜɜɟɞɟɧɢɹ ɤɥɚɫɫɢɱɟɫɤɨɝɨ ɨɩɪɟɞɟɥɟɧɢɹ ɜɟɪɨɹɬɧɨɫɬɢ ɧɟɨɛɯɨɞɢɦɨ ɩɨɡɧɚɤɨɦɢɬɶɫɹ ɫ ɧɟɤɨɬɨɪɵɦɢ ɩɨɧɹɬɢɹɦɢ. ɋɨɛɵɬɢɹ ɛɵɜɚɸɬ ɫɨɫɬɚɜɧɵɦɢ (ɧɚɩɪɢɦɟɪ, ɜɵɩɚɞɟɧɢɟ ɩɪɢ ɛɪɨɫɚɧɢɢ ɤɨɫɬɢ ɧɟ ɦɟɧɟɟ 5 ɨɱɤɨɜ) ɢ ɷɥɟɦɟɧɬɚɪɧɵɦɢ (ɜɵɩɚɞɟɧɢɟ 5 ɨɱɤɨɜ). ɗɥɟɦɟɧɬɚɪɧɵɟ ɫɨɛɵɬɢɹ ɧɟɥɶɡɹ ɪɚɡɥɨɠɢɬɶ ɧɚ ɛɨɥɟɟ ɩɪɨɫɬɵɟ, ɚ ɫɨɫɬɚɜɧɵɟ ɦɨɠɧɨ ɪɚɡɥɨɠɢɬɶ ɧɚ ɷɥɟɦɟɧɬɚɪɧɵɟ. ȼ ɪɚɧɟɟ
176
ɪɚɫɫɦɨɬɪɟɧɧɨɦ ɩɪɢɦɟɪɟ («ɧɟ ɦɟɧɟɟ 5 ɨɱɤɨɜ») ɫɨɛɵɬɢɟ ɪɚɡɥɨɠɢɦɨ ɧɚ ɞɜɟ ɷɥɟɦɟɧɬɚɪɧɵɯ: ɜɵɩɚɞɟɧɢɟ 5 ɨɱɤɨɜ, ɜɵɩɚɞɟɧɢɟ 6 ɨɱɤɨɜ. Ʉɫɬɚɬɢ, ɨɛɚ ɷɬɢɯ ɷɥɟɦɟɧɬɚɪɧɵɯ ɫɨɛɵɬɢɹ ɛɥɚɝɨɩɪɢɹɬɫɬɜɭɸɬ ɧɚɲɟɦɭ ɫɨɛɵɬɢɸ. Ɉɛɪɚɬɢɦ ɜɧɢɦɚɧɢɟ ɢ ɧɚ ɬɨ, ɱɬɨ ɜ ɫɥɭɱɚɟ ɛɪɨɫɚɧɢɹ ɤɨɫɬɢ ɜɫɟ ɲɟɫɬɶ ɷɥɟɦɟɧɬɚɪɧɵɯ ɫɨɛɵɬɢɣ (ɜɵɩɚɞɟɧɢɟ ɤɚɠɞɨɣ ɢɡ ɲɟɫɬɢ ɝɪɚɧɟɣ) ɪɚɜɧɨɜɨɡɦɨɠɧɵ, ɟɫɥɢ ɤɨɫɬɶ ɩɪɚɜɢɥɶɧɚɹ, ɢ, ɟɫɬɟɫɬɜɟɧɧɨ, ɧɟɫɨɜɦɟɫɬɧɵ, ɬ.ɟ. ɧɢɤɚɤɢɟ ɞɜɚ ɢɡ ɧɢɯ ɧɟ ɦɨɝɭɬ ɩɨɹɜɢɬɶɫɹ ɩɪɢ ɨɞɧɨɦ ɢɫɩɵɬɚɧɢɢ.
ɉɨ ɨɩɪɟɞɟɥɟɧɢɸ, ɤɥɚɫɫɢɱɟɫɤɨɣ ɜɟɪɨɹɬɧɨɫɬɶɸ ɫɨɛɵɬɢɹ Ⱥ ɧɚɡɵɜɚɟɬɫɹ ɨɬɧɨɲɟɧɢɟ ɱɢɫɥɚ ɪɚɜɧɨɜɨɡɦɨɠɧɵɯ ɷɥɟɦɟɧɬɚɪɧɵɯ ɫɨɛɵɬɢɣ, ɛɥɚɝɨɩɪɢɹɬɫɬɜɭɸɳɢɯ ɫɨɛɵɬɢɸ Ⱥ, ɤ ɨɛɳɟɦɭ ɱɢɫɥɭ ɪɚɜɧɨɜɨɡɦɨɠɧɵɯ ɷɥɟɦɟɧɬɚɪɧɵɯ ɫɨɛɵɬɢɣ:
ȼ ɧɚɲɟɦ ɩɪɢɦɟɪɟ: P(A) 2 1 .
63
ɉɭɫɬɶ ɤɨɦɩɥɟɤɫ ɭɫɥɨɜɢɣ (ɢɫɩɵɬɚɧɢɟ) ɫɨɫɬɨɢɬ ɜ ɩɨɞɛɪɚɫɵɜɚɧɢɢ ɞɜɭɯ ɢɞɟɚɥɶɧɵɯ ɤɨɫɬɟɣ. Ʉɚɤɨɜɚ ɜɟɪɨɹɬɧɨɫɬɶ ɬɨɝɨ, ɱɬɨ ɲɟɫɬɟɪɤɚ ɜɵɩɚɞɚɟɬ ɞɜɚ ɪɚɡɚ (Ⱥ)? Ɉɞɢɧ ɪɚɡ (ȼ)? ɏɨɬɹ ɛɵ ɨɞɢɧ ɪɚɡ (ɋ)?
Ɍɟɩɟɪɶ ɢɦɟɟɬɫɹ ɜɫɟɝɨ 36 ɷɥɟɦɟɧɬɚɪɧɵɯ ɪɚɜɧɨɜɨɡɦɨɠɧɵɯ ɫɨɛɵɬɢɣ: ɤɚɠɞɚɹ ɢɡ ɝɪɚɧɟɣ ɨɞɧɨɣ ɤɨɫɬɢ ɜɵɩɚɞɚɟɬ ɫ ɤɚɠɞɨɣ ɝɪɚɧɶɸ ɞɪɭɝɨɣ.
ɋɨɛɵɬɢɸ Ⱥ - «ɜɵɩɚɞɚɟɬ ɞɜɚ ɪɚɡɚ» - ɛɥɚɝɨɩɪɢɹɬɫɬɜɭɟɬ ɬɨɥɶɤɨ ɨɞɧɨ ɷɥɟɦɟɧɬɚɪɧɨɟ ɫɨɛɵɬɢɟ, ɫɥɟɞɨɜɚɬɟɥɶɧɨ, P(A) 36 . ɋɨɛɵɬɢɸ ȼ ɛɥɚɝɨɩɪɢɹɬɫɬɜɭɸɬ ɬɚɤɢɟ ɷɥɟɦɟɧɬɚɪɧɵɟ: ɧɚ ɩɟɪɜɨɣ ɤɨɫɬɢ ɲɟɫɬɟɪɤɚ, ɚ ɧɚ ɞɪɭɝɨɣ ɧɟ ɲɟɫɬɟɪɤɚ (ɢɯ 5), ɧɚ ɩɟɪɜɨɣ ɧɟ ɲɟɫɬɟɪɤɚ, ɧɚ ɜɬɨɪɨɣ -
ɲɟɫɬɟɪɤɚ (ɢɯ ɬɨɠɟ 5), ɬ.ɟ. P(B) |
10 |
|
|
5 |
. ɋɨɛɵɬɢɸ ɋ ɛɥɚɝɨɩɪɢɹɬɫɬɜɭɸɬ ɜɫɟ ɬɟ, ɤɨɬɨɪɵɟ |
||||
|
|
||||||||
|
|
|
36 |
|
18 |
|
|||
ɛɥɚɝɨɩɪɢɹɬɫɬɜɭɸɬ ȼ ɩɥɸɫ ɟɳɟ ɨɞɧɨ: |
|
ɲɟɫɬɟɪɤɚ ɧɚ ɩɟɪɜɨɣ ɢ ɧɚ ɜɬɨɪɨɣ ɤɢɫɬɢ. Ɍɟɩɟɪɶ |
|||||||
P(C) |
10 1 |
|
11 |
. |
|
|
|
|
|
|
|
|
|
|
|
|
|||
36 |
|
36 |
|
|
|
|
|
|
|
Ɉɬɦɟɬɢɦ, ɱɬɨ ɟɫɥɢ ɟɫɬɶ ɞɜɚ ɧɟɡɚɜɢɫɢɦɵɯ ɫɨɛɵɬɢɹ Ⱥ ɢ ȼ (ɬ.ɟ. ɞɜɚ ɬɚɤɢɯ ɫɨɛɵɬɢɹ, ɱɬɨ ɨɫɭɳɟɫɬɜɥɟɧɢɟ ɨɞɧɨɝɨ ɢɡ ɧɢɯ ɧɢɤɚɤ ɧɟ ɫɤɚɡɵɜɚɟɬɫɹ ɧɚ ɨɫɭɳɟɫɬɜɥɟɧɢɢ ɞɪɭɝɨɝɨ), ɬɨ ɜɟɪɨɹɬɧɨɫɬɶ ɫɨɜɦɟɫɬɧɨɝɨ ɨɫɭɳɟɫɬɜɥɟɧɢɹ Ⱥ ɢ ȼ ɪɚɜɧɚ Ɋ (Ⱥ) • Ɋ (ȼ), Ɍɚɤ, ɜ ɪɚɫɫɦɨɬɪɟɧɧɨɦ ɩɪɢɦɟɪɟ ɜɵɩɚɞɟɧɢɟ ɲɟɫɬɟɪɤɢ ɧɚ ɨɞɧɨɣ ɤɨɫɬɢ ɧɢɤɚɤ ɧɟ ɜɥɢɹɟɬ ɧɚ ɜɵɩɚɞɟɧɢɟ ɬɨɣ ɢɥɢ ɢɧɨɣ
[250]
ɝɪɚɧɢ ɧɚ ɞɪɭɝɨɣ ɤɨɫɬɢ, ɚ ɩɨɫɤɨɥɶɤɭ ɜɟɪɨɹɬɧɨɫɬɶ ɜɵɩɚɞɟɧɢɹ ɲɟɫɬɟɪɤɢ ɧɚ ɨɞɧɨɣ ɤɨɫɬɢ ɪɚɜɧɚ
1 |
ɬɨ ɜɵɩɚɞɟɧɢɟ ɲɟɫɬɟɪɤɢ ɧɚ ɞɜɭɯ ɤɨɫɬɹɯ ɨɞɧɨɜɪɟɦɟɧɧɨ ɪɚɜɧɨ |
1 |
u |
1 |
|
1 |
. |
|
|
|
|
||||
6 |
6 |
6 |
36 |
|
|||
Ʉɚɤ ɜɢɞɢɦ, ɷɬɨɬ ɪɟɡɭɥɶɬɚɬ ɫɨɜɩɚɞɚɟɬ ɫ ɩɨɥɭɱɟɧɧɵɦ ɪɚɧɟɟ ɢɡ ɞɪɭɝɢɯ ɫɨɨɛɪɚɠɟɧɢɣ.
ɍɩɪɚɠɧɟɧɢɟ 98.
1.ɉɨɞɛɪɚɫɵɜɚɸɬɫɹ ɞɜɟ ɦɨɧɟɬɵ. ɇɚɣɬɢ ɜɟɪɨɹɬɧɨɫɬɶ ɜɵɩɚɞɟɧɢɹ ɝɟɪɛɚ ɯɨɬɹ ɛɵ ɧɚ ɨɞɧɨɣ ɢɡ ɧɢɯ. Ɉɬɜɟɬ: 34 .
2.Ɍɨ ɠɟ ɜ ɫɥɭɱɚɟ ɬɪɟɯ ɦɨɧɟɬ. Ɉɬɜɟɬ: 78 .
3.ɉɨɞɛɪɚɫɵɜɚɸɬɫɹ 3 ɦɨɧɟɬɵ. Ʉɚɤɨɜɚ ɜɟɪɨɹɬɧɨɫɬɶ ɜɵɩɚɞɟɧɢɹ ɝɟɪɛɚ ɧɚ ɜɫɟɯ ɦɨɧɟɬɚɯ. Ɉɬɜɟɬ: 18 .
4.ȼ ɩɚɪɬɢɢ ɢɡ 100 ɥɚɦɩ 6 ɛɪɚɤɨɜɚɧɧɵɯ. Ʉɚɤɨɜɚ ɜɟɪɨɹɬɧɨɫɬɶ, ɱɬɨ ɢɡ ɞɜɭɯ ɥɚɦɩ, ɜɡɹɬɵɯ ɧɚ ɢɫɩɵɬɚɧɢɟ, ɨɛɟ ɨɤɚɠɭɬɫɹ ɧɟɢɫɩɪɚɜɧɵɦɢ?
ɍɤɚɡɚɧɢɟ: ɧɚɣɬɢ ɜɟɪɨɹɬɧɨɫɬɶ, ɱɬɨ 1-ɹ ɥɚɦɩɚ ɛɪɚɤɨɜɚɧɧɚɹ; ɩɨɫɥɟ ɩɪɨɜɟɪɤɢ 1-ɣ ɥɚɦɩɵ ɨɫɬɚɟɬɫɹ 99 ɥɚɦɩ - ɧɚɣɬɢ ɜɟɪɨɹɬɧɨɫɬɶ ɬɨɝɨ, ɱɬɨ 2-ɹ ɥɚɦɩɚ ɛɪɚɤɨɜɚɧɧɚɹ, ɜɨɫɩɨɥɶɡɨɜɚɬɶɫɹ
ɮɨɪɦɭɥɨɣ ɩɟɪɟɦɧɨɠɟɧɢɹ ɜɟɪɨɹɬɧɨɫɬɟɣ ɧɟɡɚɜɢɫɢɦɵɯ ɫɨɛɵɬɢɣ. Ɉɬɜɟɬ: 6u5 | 0.003 100u99
ɇɟɞɨɫɬɚɬɤɨɦ ɤɥɚɫɫɢɱɟɫɤɨɝɨ ɨɩɪɟɞɟɥɟɧɢɹ, ɨɱɟɜɢɞɧɨ, ɹɜɥɹɟɬɫɹ ɞɢɫɤɪɟɬɧɨɫɬɶ ɩɪɨɫɬɪɚɧɫɬɜɚ ɷɥɟɦɟɧɬɚɪɧɵɯ ɫɨɛɵɬɢɣ. ɗɬɨɬ ɧɟɞɨɫɬɚɬɨɤ ɭɫɬɪɚɧɹɟɬɫɹ ɜ ɬɚɤ ɧɚɡɵɜɚɟɦɨɦ ɝɟɨɦɟɬɪɢɱɟɫɤɨɦ ɩɨɧɹɬɢɢ ɜɟɪɨɹɬɧɨɫɬɢ. ɉɭɫɬɶ ɧɚ ɩɥɨɫɤɨɫɬɢ ɢɦɟɟɬɫɹ ɧɟɤɨɬɨɪɚɹ ɨɛɥɚɫɬɶ G,
177
ɜɧɭɬɪɢ ɤɨɬɨɪɨɣ ɫɨɞɟɪɠɢɬɫɹ ɨɛɥɚɫɬɶ g. Ʉɨɦɩɥɟɤɫ ɭɫɥɨɜɢɣ ɡɚɤɥɸɱɚɟɬɫɹ ɜ ɛɪɨɫɚɧɢɢ ɧɚɭɞɚɱɭ ɬɨɱɤɢ. ɑɟɦɭ ɪɚɜɧɚ ɜɟɪɨɹɬɧɨɫɬɶ ɩɨɩɚɞɚɧɢɹ ɬɨɱɤɢ ɜ g?
ɉɪɨɫɬɪɚɧɫɬɜɨ ɷɥɟɦɟɧɬɚɪɧɵɯ ɫɨɛɵɬɢɣ ɬɟɩɟɪɶ ɧɟɩɪɟɪɵɜɧɨ, ɨɧɨ ɫɨɫɬɨɢɬ ɢɡ ɛɟɫɤɨɧɟɱɧɨɝɨ ɱɢɫɥɚ ɬɨɱɟɤ. ɋɨɜɨɤɭɩɧɨɫɬɶ ɬɨɱɟɤ ɨɛɥɚɫɬɢ g ɨɛɪɚɡɭɟɬ ɦɧɨɠɟɫɬɜɨ ɷɥɟɦɟɧɬɚɪɧɵɯ ɫɨɛɵɬɢɣ, ɛɥɚɝɨɩɪɢɹɬɫɬɜɭɸɳɢɯ ɨɛɫɭɠɞɚɟɦɨɦɭ. Ɍɚɤ ɤɚɤ ɭɫɥɨɜɢɟ ɪɚɜɧɨɜɨɡɦɨɠɧɨɫɬɢ ɫɨɯɪɚɧɹɟɬɫɹ ɢ ɡɞɟɫɶ, ɬɨ ɟɫɬɟɫɬɜɟɧɧɨ ɨɩɪɟɞɟɥɢɬɶ ɜɟɪɨɹɬɧɨɫɬɶ ɫɨɛɵɬɢɹ ɤɚɤ ɨɬɧɨɲɟɧɢɟ ɦɟɪ ɨɛɥɚɫɬɟɣ (ɜ ɞɚɧɧɨɦ ɫɥɭɱɚɟ - ɩɥɨɳɚɞɟɣ, ɧɨ ɷɬɨ ɦɨɠɟɬ ɛɵɬɶ ɨɬɧɨɲɟɧɢɟ ɞɥɢɧ ɨɬɪɟɡɤɨɜ, ɨɛɴɟɦɨɜ ɢ ɬ.ɞ.)
ɉɪɨɫɬɪɚɧɫɬɜɨ ɷɥɟɦɟɧɬɚɪɧɵɯ ɫɨɛɵɬɢɣ ɢɦɟɟɬ ɧɭɥɟɜɭɸ ɜɟɪɨɹɬɧɨɫɬɶ (ɦɟɪɚ ɬɨɱɤɢ ɟɫɬɶ ɧɭɥɶ). ȼ ɬɨ ɠɟ ɜɪɟɦɹ ɥɸɛɨɟ ɢɡ ɷɥɟɦɟɧɬɚɪɧɵɯ ɫɨɛɵɬɢɣ ɹɜɥɹɟɬɫɹ ɜɨɡɦɨɠɧɵɦ. Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, ɫɭɳɟɫɬɜɭɸɬ ɫɨɛɵɬɢɹ, ɨɫɭɳɟɫɬɜɥɟɧɢɟ ɤɨɬɨɪɵɯ ɧɚ ɨɩɵɬɟ ɜɨɡɦɨɠɧɨ, ɯɨɬɹ ɨɧɢ ɢɦɟɸɬ ɧɭɥɟɜɭɸ ɜɟɪɨɹɬɧɨɫɬɶ.
Ʉɚɤ ɭɠɟ ɨɬɦɟɱɚɥɨɫɶ, ɝɟɨɦɟɬɪɢɱɟɫɤɨɟ ɨɩɪɟɞɟɥɟɧɢɟ, ɤɚɤ ɢ ɤɥɚɫɫɢɱɟɫɤɨɟ, ɬɪɟɛɭɟɬ ɜɵɩɨɥɧɟɧɢɹ ɭɫɥɨɜɢɹ ɪɚɜɧɨɜɨɡɦɨɠɧɨɫɬɢ. ȼ ɷɬɨɦ ɟɝɨ ɫɭɳɟɫɬɜɟɧɧɵɣ ɧɟɞɨɫɬɚɬɨɤ. ɗɬɨɬ ɧɟɞɨɫɬɚɬɨɤ ɩɪɟɨɞɨɥɟɜɚɟɬɫɹ ɜ ɬɚɤ ɧɚɡɵɜɚɟɦɨɦ ɚɤɫɢɨɦɚɬɢɱɟɫɤɨɦ ɨɩɪɟɞɟɥɟɧɢɢ, ɢɡɥɨɠɟɧɢɟ ɤɨɬɨɪɨɝɨ ɜɵɯɨɞɢɬ ɢɡ ɪɚɦɤɢ ɞɚɧɧɨɣ ɤɧɢɝɢ4.
ɉɪɢɥɨɠɟɧɢɟ 2.
ɋɍɆɆɕ ɂ ɇȿɄɈɌɈɊɕȿ ɁȺȾȺɑɂ ɇȺ ɋɍɆɆɂɊɈȼȺɇɂȿ
Ɉ ɫɭɦɦɚɯ. Ⱦɥɹ ɡɚɩɢɫɢ ɫɭɦɦɵ ɪɹɞɚ ɱɢɫɟɥ ɢɫɩɨɥɶɡɭɟɬɫɹ ɝɪɟɱɟɫɤɚɹ ɩɪɨɩɢɫɧɚɹ ɛɭɤɜɚ
|
|
|
4 |
|
(ɫɢɝɦɚ), ɇɚɩɪɢɦɟɪ, A1+A2+A3+A4 |
ɡɚɩɢɫɵɜɚɟɬɫɹ ¦Ai . |
ȿɫɥɢ ɭ ɧɚɫ n ɫɥɚɝɚɟɦɵɯ, ɬɨ ɢɯ |
||
|
|
|
i 1 |
|
ɡɚɩɢɫɵɜɚɸɬ ɫɥɟɞɭɸɳɢɦ |
|
|
|
|
[251] |
|
|
|
|
|
|
n |
|
|
ɨɛɪɚɡɨɦ A1 A2 ...An |
¦Ai |
ɢ ɱɢɬɚɸɬ: «ɫɭɦɦɚ Ai |
, ɝɞɟ i ɢɡɦɟɧɹɟɬɫɹ ɞɨɬ 1 ɞɨ n». ɗɬɨ |
|
|
|
i 1 |
|
|
ɨɛɨɡɧɚɱɟɧɢɟ ɫɭɳɟɫɬɜɟɧɧɨ ɨɛɥɟɝɱɚɟɬ ɡɚɩɢɫɶ ɫɭɦɦ. |
|
|||
ɂɡ ɨɩɪɟɞɟɥɟɧɢɹ ¦ ɫɥɟɞɭɟɬ, ɱɬɨ |
|
|||
n |
n |
n |
|
|
¦ Dai Ebi J D¦ai E¦bi nJ |
|
|||
i 1 |
i 1 |
i |
1 |
|
Ⱦɟɣɫɬɜɢɬɟɥɶɧɨ, |
|
|
|
|
n |
|
|
|
|
¦ Dai Ebi J Da1 Eb1 J Da2 Eb2 J |
|
|||
i 1 |
|
|
|
|
... Dan Ebn J |
D a1 a2 ... an |
|
||
E b1 b2 ... bn nJ |
n |
n |
|
|
D¦ai E¦bi nJ |
|
|||
|
|
i 1 |
i 1 |
|
Ɉɬɦɟɬɢɦ, ɱɬɨ ɪɟɡɭɥɶɬɚɬ ɫɭɦɦɢɪɨɜɚɧɢɹ ɧɟ ɡɚɜɢɫɢɬ ɨɬ ɬɨɝɨ, ɤɚɤ ɨɛɨɡɧɚɱɟɧ ɢɧɞɟɤɫ ɫɭɦɦɢɪɨɜɚɧɢɹ:
nn
¦Ai |
¦AS A1 A2 ... An |
i 1 |
i 1 |
Ɂɚ ɡɧɚɤ ɫɭɦɦɵ ɩɨ i , ɧɚɩɪɢɦɟɪ, ɦɨɠɧɨ ɜɵɧɨɫɢɬɶ ɥɸɛɨɟ ɜɵɪɚɠɟɧɢɟ, ɧɟ ɫɨɞɟɪɠɚɳɟɟ i , ɞɚɠɟ ɟɫɥɢ ɨɧɨ ɫɨɞɟɪɠɢɬ ɞɪɭɝɢɟ ɢɧɞɟɤɫɵ ɢɥɢ ɫɭɦɦɵ.
ɇɚɩɪɢɦɟɪ,
nn
¦Ai Bj |
Bj ¦Ai |
i 1 |
i 1 |
4 Ʌɸɛɨɡɧɚɬɟɥɶɧɨɝɨ ɱɢɬɚɬɟɥɹ ɦɵ ɨɬɫɵɥɚɟɦ ɤ ɤɧɢɝɟ: Ʉɨɥɦɨɝɨɪɨɜ Ⱥ. ɇ. Ɉɫɧɨɜɧɵɟ ɩɨɧɹɬɢɹ ɬɟɨɪɢɢ ɜɟɪɨɹɬɧɨɫɬɟɣ. Ɇ., 1974; ɋɦ. ɬɚɤɠɟ: ɉɭɝɚɱɟɜ ȼ. ɋ. Ɍɟɨɪɢɹ ɜɟɪɨɹɬɧɨɫɬɢ ɢ ɦɚɬɟɦɚɬɢɱɟɫɤɚɹ ɫɬɚɬɢɫɬɢɤɚ. Ɇ., 1979.
178
n § |
|
k |
|
· |
§ |
|
k |
|
· n |
|
¦¨¨Ai |
¦Cj Bj ¸¸ |
¨¨ |
¦C j Bj ¸¸¦Ai |
|||||||
i 1 © |
|
j 1 |
¹ © j 1 |
|
¹i 1 |
|||||
Ɉ ɫɭɦɦɢɪɨɜɚɧɢɢ ɫɬɟɩɟɧɟɣ ɧɚɬɭɪɚɥɶɧɵɯ ɱɢɫɟɥ. |
||||||||||
Ɋɚɫɫɦɨɬɪɢɦ |
|
|
|
|
|
|
|
|||
Snk |
|
|
|
|
|
|
|
|
|
n |
1k |
2k |
... nk |
¦ik |
|||||||
|
|
|
|
|
|
|
|
|
i 1 |
|
ɉɪɢ k |
|
1 |
|
|
|
1 n |
|
|||
S 1 |
1 2 ... n |
|
|
n |
||||||
|
|
|
||||||||
n |
|
|
|
|
|
2 |
|
|
||
|
|
|
|
|
|
|
|
|||
— ɨɛɵɱɧɚɹ ɫɭɦɦɚ ɱɥɟɧɨɜ ɚɪɢɮɦɟɬɢɱɟɫɤɨɣ ɩɪɨɝɪɟɫɫɢɢ. |
||||||||||
Ɉɫɨɛɵɣ ɢɧɬɟɪɟɫ ɩɪɟɞɫɬɚɜɥɹɟɬ ɞɥɹ ɧɚɫ ɫɥɭɱɚɣ k=2: |
||||||||||
S 2 |
12 22 |
... n2 |
|
|
||||||
n |
|
|
|
|
|
|
|
|
|
|
ɉɨɤɚɠɟɦ ɤɚɤ ɧɚɣɬɢ S |
2 |
|
||||||||
|
|
|
|
|
|
|
|
n |
||
Ɋɚɫɫɦɨɬɪɢɦ |
|
|
|
|
|
|
|
|||
(n 1)3 n3 |
3n2 3n 1 |
|||||||||
[252] |
|
|
|
|
|
|
|
|
|
|
n3 n 1 3 |
3 n 1 2 3 n 1 1 |
|||||||||
................................................. |
||||||||||
33 23 |
3 22 3 2 1 |
|
|
|||||||
23 13 |
|
3 12 3 1 1 |
|
|
||||||
ɉɪɨɫɭɦɦɢɪɭɟɦ ɩɨɱɥɟɧɧɨ ɜɫɟ ɪɚɜɟɧɫɬɜɚ: |
||||||||||
n 1 3 1 3 Sn2 3 Sn1 n |
||||||||||
Ɍɚɤ ɤɚɤ S 1 |
ɧɚɦ ɢɡɜɟɫɬɧɨ, ɬɨ ɩɨɫɥɟɞɧɟɟ ɭɪɚɜɧɟɧɢɟ ɫɨɞɟɪɠɢɬ ɥɢɲɶ ɨɞɧɭ ɧɟɢɡɜɟɫɬɧɭɸ |
|||||||||
|
|
2 |
n |
|
|
|
|
|
|
|
ɜɟɥɢɱɢɧɭ S |
. Ɋɟɲɚɹ ɟɝɨ, ɩɨɥɭɱɢɦ ɩɨɫɥɟ ɩɪɨɫɬɵɯ ɩɪɟɨɛɪɚɡɨɜɚɧɢɣ: |
|||||||||
|
|
n |
|
|
|
|
|
|
|
|
S 2 |
|
n n 1 2n 1 |
. |
|
|
|||||
|
|
|
|
|||||||
n |
|
|
|
6 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
ɋɩɪɚɜɟɞɥɢɜɨɫɬɶ ɷɬɨɝɨ ɭɬɜɟɪɠɞɟɧɢɹ ɦɨɠɟɬ ɛɵɬɶ ɩɨɤɚɡɚɧɚ ɢ ɫ ɩɨɦɨɳɶɸ ɦɟɬɨɞɚ |
||||||||||
ɦɚɬɟɦɚɬɢɱɟɫɤɨɣ ɢɧɞɭɤɰɢɢ. |
|
|
||||||||
ɇɚɣɞɟɦ S 3 |
. Ⱦɥɹ ɷɬɨɝɨ ɦɨɠɧɨ ɢɫɩɨɥɶɡɨɜɚɬɶ ɫɨɨɬɧɨɲɟɧɢɟ |
|||||||||
|
|
|
n |
|
|
|
|
|
|
|
n 1 4 1 4 n3 6 n2 4n 1,
ɜ ɫɩɪɚɜɟɞɥɢɜɨɫɬɢ ɤɨɬɨɪɨɝɨ ɩɪɨɫɬɨ ɭɛɟɞɢɬɶɫɹ. Ɂɚɩɢɫɵɜɚɹ ɟɝɨ n ɪɚɡ, ɤɚɤ ɷɬɨ ɛɵɥɨ ɜɵɲɟ
ɫɞɟɥɚɧɨ ɩɪɢ ɧɚɯɨɠɞɟɧɢɢ S |
2 , ɩɨɥɭɱɢɦ ɩɨɫɥɟ ɩɨɱɥɟɧɧɨɝɨ ɫɭɦɦɢɪɨɜɚɧɢɹ |
|||||||||||||||||
|
|
|
|
|
|
|
|
|
|
|
n |
|
|
|
|
|
|
|
n 1 4 1 4 Sn3 6 Sn2 4Sn1 n. |
|
|
|
|
||||||||||||||
Ɉɬɫɸɞɚ |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
||||
3 |
ªn n 1 º2 |
|
|
|
|
|
|
|
|
|
||||||||
Sn |
|
« |
|
|
|
» . |
|
|
|
|
|
|
|
|
|
|
||
|
|
2 |
|
|
|
|
|
|
|
|
|
|
||||||
|
|
|
¬ |
|
|
¼ |
|
|
|
|
|
|
|
|
|
|
|
|
ɉɪɢ ɪɚɫɫɦɨɬɪɟɧɢɢ ɤɨɷɮɮɢɰɢɟɧɬɚ ɪɚɧɝɨɜɨɣ ɤɨɪɪɟɥɹɰɢɢ ɋɩɢɪɦɟɧɚ ɧɟɨɛɯɨɞɢɦɨ ɡɧɚɧɢɟ |
||||||||||||||||||
ɫɭɦɦɵ |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
n |
§ |
|
n |
1 |
·2 |
|
n |
2 |
|
|
n |
n n 1 2 |
||||||
¦ |
¨i |
|
|
|
¸ |
|
¦i |
|
n 1 ¦i |
|
|
|
|
|
||||
|
|
|
|
|
|
|
|
|||||||||||
i 1 |
© |
|
|
2 |
¹ |
|
i 1 |
|
|
|
i 1 |
|
4 |
|
|
|
||
|
n n 1 2n 1 |
|
n n 1 2 |
|
n n 1 2 |
|
n3 n |
. |
||||||||||
|
|
|
|
|
||||||||||||||
|
|
|
|
6 |
|
|
|
|
|
2 |
4 |
|
12 |
|
||||
ɚ ɬɚɤɠɟ ɫɭɦɦɵ ɤɜɚɞɪɚɬɨɜ n ɧɟɱɟɬɧɵɯ ɱɢɫɟɥ:
179
|
|
|
|
|
n2 |
12 |
32 |
|
52 |
... 2n 1 2 |
|
|||||||||||||||||||||||||||
S |
|
|||||||||||||||||||||||||||||||||||||
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
n( |
4n2 1 |
|
|
|
|
|
|
|
|
|
|||
ɉɨɤɚɠɟɦ, ɱɬɨ S |
|
2 |
ɫ ɩɨɦɨɳɶɸ ɦɟɬɨɞɚ ɦɚɬɟɦɚɬɢɱɟɫɤɨɣ ɢɧɞɭɤɰɢɢ. |
|||||||||||||||||||||||||||||||||||
1 |
|
|
|
3 |
|
|||||||||||||||||||||||||||||||||
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
||||
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
1 3 |
|
|
|
|
|
||
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
1 — ɩɨ ɨɩɪɟɞɟɥɟɧɢɸ ɢ |
1 – ɩɨ ɮɨɪɦɭɥɟ. |
||||||||||||||||||
Ⱦɥɹ n |
|
|
|
|
1 S |
|
2 |
|
|
|||||||||||||||||||||||||||||
|
|
|
|
1 |
|
|
|
|
|
|||||||||||||||||||||||||||||
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
3 |
|
|
|
|
|
|||||
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
2 15 |
|
|||||
|
|
|
|
|
|
|
|
|
2 : |
|
|
|
|
1 3 |
|
|
10 - ɩɨ ɨɩɪɟɞɟɥɟɧɢɸ ɢ |
|
||||||||||||||||||||
( Ⱦɥɹ |
n |
|
|
S |
2 |
|
|
10 ɩɨ ɮɨɪɦɭɥɟ. |
||||||||||||||||||||||||||||||
|
|
|
|
|
|
|||||||||||||||||||||||||||||||||
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
2 |
|
|
|
|
|
|
|
|
|
3 |
|
||||||||
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
||||||||
[253] |
|
|
|
|
|
|
|
|
|
k 4k 2 1 |
|
|
|
|
|
|
|
|
|
|
|
k 1 4k 2 8k 3 |
||||||||||||||||
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
2 |
||||||||||||||||
|
|
|
|
|
|
|
|
|
|
|
|
2 |
|
|
|
|
|
|
||||||||||||||||||||
ɉɭɫɬɶ S k |
|
|
|
|
|
|
|
|
|
|
|
|
, ɩɨɤɚɠɟɦ, ɱɬɨ ɩɪɢ ɷɬɨɦ S k 1 |
|
||||||||||||||||||||||||
|
|
|
|
|
|
|
3 |
|
|
3 |
||||||||||||||||||||||||||||
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
k 4k 2 1 |
|
|
|
|
|
|
|||||||
|
|
|
|
k 21 |
|
|
|
k 2 2k 1 2 |
|
|
|
|||||||||||||||||||||||||||
S |
S |
|
||||||||||||||||||||||||||||||||||||
|
|
|
|
|
|
|
|
|||||||||||||||||||||||||||||||
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
3 |
|
|
|
|
|
|
|
|
|
|
|
2k 1 2 |
|
|
|
k 1 4k 2 8k 3 |
. |
|
||||||||||||||||||||||||||||||||
|
|
|
|
|
||||||||||||||||||||||||||||||||||
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
3 |
|
|
|
|
|
|
|
|
|
|
|
|||
ɍɩɪɚɠɧɟɧɢɟ 99. ɇɚɣɬɢ ɫɭɦɦɭ ɤɜɚɞɪɚɬɨɜ n ɱɟɬɧɵɯ ɱɢɫɟɥ |
||||||||||||||||||||||||||||||||||||||
|
|
2 |
22 42 62 ... 2n 2 |
|
|
|
|
|
|
|
|
|
||||||||||||||||||||||||||
|
|
|
|
|
|
|
|
|
|
|||||||||||||||||||||||||||||
|
|
|
|
|
|
|
|
|
|
|
|
|
||||||||||||||||||||||||||
|
S n |
|
|
|
|
|
|
|
|
|
||||||||||||||||||||||||||||
Ɉɬɜɟɬ: |
2 |
n n 1 2n 1 |
|
|
|
|
|
|
|
|
|
|||||||||||||||||||||||||||
|
|
|
|
|
|
|
|
|
|
|||||||||||||||||||||||||||||
|
|
|
|
|
|
|
3 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
n 1 2 2n2 4n 1 |
|||||
ɍɩɪɚɠɧɟɧɢɟ 100. ɉɨɤɚɡɚɬɶ, ɱɬɨ 13 33 ... 2n 1 3 |
||||||||||||||||||||||||||||||||||||||
ȼ ɧɟɤɨɬɨɪɵɯ ɫɥɭɱɚɹɯ ɜɟɥɢɱɢɧɵ ɦɨɝɭɬ ɨɤɚɡɚɬɶɫɹ ɫɧɚɛɠɟɧɧɵɦɢ ɞɜɭɦɹ ɢɧɞɟɤɫɚɦɢ. ɇɚɩɪɢɦɟɪ, ɜ ɤɨɪɪɟɥɹɰɢɨɧɧɨɣ ɬɚɛɥɢɰɟ 15 (§ 1 ɝɥɚɜɵ II) Nij - ɱɢɫɥɨ ɢɧɞɢɜɢɞɨɜ, ɭ ɤɨɬɨɪɵɯ ɧɟɤɨɬɨɪɵɣ ɩɪɢɡɧɚɤ ɏ ɩɪɢɧɢɦɚɟɬ ɡɧɚɱɟɧɢɟ xi ɚ ɞɪɭɝɨɣ ɩɪɢɡɧɚɤ Y ɡɧɚɱɟɧɢɟ yi. Ɉɱɟɜɢɞɧɨ,
k
¦Nij , ɝɞɟ k - ɱɢɫɥɨ ɡɧɚɱɟɧɢɣ ɩɪɢɡɧɚɤɚ ɏ, ɞɚɟɬ ɫɭɦɦɭ ɷɥɟɦɟɧɬɨɜ j-ɨɝɨ ɫɬɨɥɛɰɚ ɬɚɛɥɢɰɵ, ɬ.ɟ.
i 1
ɱɢɫɥɨ ɢɧɞɢɜɢɞɨɜ, ɭ ɤɨɬɨɪɵɯ Y=yi (ɧɟɡɚɜɢɫɢɦɨ ɨɬ ɬɨɝɨ, ɤɚɤɨɜɨ ɡɧɚɱɟɧɢɟ ɩɪɢɡɧɚɤɚ ɏ).
l
Ⱥɧɚɥɨɝɢɱɧɨ ¦Nij , ɝɞɟ l - ɱɢɫɥɨ ɡɧɚɱɟɧɢɣ ɩɪɢɡɧɚɤɚ Y, ɞɚɟɬ ɫɭɦɦɭ ɷɥɟɦɟɧɬɨɜ i-ɨɣ
j 1
ɫɬɪɨɤɢ ɦɚɬɪɢɰɵ, ɬ.ɟ. ɱɢɫɥɨ ɢɧɞɢɜɢɞɨɜ, ɭ ɤɨɬɨɪɵɯ X=xi (ɧɟɡɚɜɢɫɢɦɨ ɨɬ ɬɨɝɨ, ɤɚɤɨɜɨ ɡɧɚɱɟɧɢɟ ɩɪɢɡɧɚɤɚ Y).
ɋɭɦɦɢɪɭɹ ɩɨ ɨɛɨɢɦ ɢɧɞɟɤɫɚɦ, ɦɵ, ɨɱɟɜɢɞɧɨ, ɩɨɥɭɱɢɦ ɨɛɳɟɟ ɱɢɫɥɨ ɧɚɛɥɸɞɟɧɢɣ (ɤɨɥɢɱɟɫɬɜɨ ɢɧɞɢɜɢɞɨɜ ɨɛɫɥɟɞɭɟɦɨɣ ɨɛɳɧɨɫɬɢ), ɬ.ɟ.
kl
N¦¦Nij
i 1 j 1
ɉɪɢɥɨɠɟɧɢɟ 3.
ɋɌȺɌɂɋɌɂɑȿɋɄɂȿ ɌȺȻɅɂɐɕ
ɉɪɢɜɟɞɟɧɧɵɟ ɧɢɠɟ ɬɚɛɥɢɰɵ ɱɚɫɬɢɱɧɨ ɡɚɢɦɫɬɜɨɜɚɧɵ ɢɡ ɞɪɭɝɢɯ ɢɡɞɚɧɢɣ, ɱɚɫɬɢɱɧɨ ɪɚɫɫɱɢɬɚɧɵ ɧɚ «ɗɥɟɤɬɪɨɧɢɤɟ ȻɁ - 21» ɩɨ ɫɨɫɬɚɜɥɟɧɧɵɦ ɚɜɬɨɪɚɦɢ ɩɪɨɝɪɚɦɦɚɦ. ɉɪɢ ɩɨɞɛɨɪɟ ɬɚɛɥɢɰ ɦɵ ɪɭɤɨɜɨɞɫɬɜɨɜɚɥɢɫɶ, ɜɨ-ɩɟɪɜɵɯ, ɫɨɨɛɪɚɠɟɧɢɹɦɢ ɭɞɨɛɫɬɜɚ (ɩɪɢɜɨɞɢɦɵɟ ɜ ɧɟɤɨɬɨɪɵɯ ɢɡɞɚɧɢɹɯ ɬɚɛɥɢɰɵ ɬɪɟɛɭɸɬ ɢɧɨɝɞɚ ɩɟɪɟɫɱɟɬɚ ɩɪɢ ɩɨɥɶɡɨɜɚɧɢɢ5 - ɡɞɟɫɶ ɷɬɨɬ ɧɟɞɨɫɬɚɬɨɤ ɭɫɬɪɚɧɟɧ); ɜɨ-ɜɬɨɪɵɯ, ɫɨɨɛɪɚɠɟɧɢɹɦɢ ɫɨɨɬɜɟɬɫɬɜɢɹ ɬɚɛɥɢɰ ɪɚɫ-
[254]
5 ɋɦ., ɧɚɩɪɢɦɟɪ: ɋɬɚɬɢɫɬɢɱɟɫɤɢɟ ɦɟɬɨɞɵ ɚɧɚɥɢɡɚ ɢɧɮɨɪɦɚɰɢɢ ɜ ɫɨɰɢɨɥɨɝɢɱɟɫɤɢɯ ɢɫɫɥɟɞɨɜɚɧɢɹɯ, Ɇ., 1979, ɬɚɛɥ. Ⱥ ɧɚ ɫ. 299.
180
ɩɪɨɫɬɪɚɧɟɧɧɵɦ ɜ ɫɨɰɢɨɥɨɝɢɱɟɫɤɢɯ ɢɫɫɥɟɞɨɜɚɧɢɹɯ ɮɨɪɦɚɦ ɩɪɟɞɫɬɚɜɥɟɧɢɹ ɢ ɤɨɥɢɱɟɫɬɜɟɧɧɵɦ ɯɚɪɚɤɬɟɪɢɫɬɢɤɚɦ ɢɧɮɨɪɦɚɰɢɢ (ɧɚɩɪɢɦɟɪ, ɬɚɛɥɢɰɚ F2 ɱɚɫɬɨ ɩɪɢɜɨɞɢɬɫɹ ɜ ɫɬɚɬɢɫɬɢɱɟɫɤɨɣ ɥɢɬɟɪɚɬɭɪɟ6 ɥɢɲɶ ɞɥɹ ɱɢɫɥɚ ɫɬɟɩɟɧɟɣ ɫɜɨɛɨɞɵ, ɧɟ ɩɪɟɜɵɲɚɸɳɢɯ 30, ɜ ɬɨ ɜɪɟɦɹ ɤɚɤ ɜ ɫɨɰɢɨɥɨɝɢɱɟɫɤɢɯ ɢɫɫɥɟɞɨɜɚɧɢɹɯ ɧɟɪɟɞɤɢ ɬɚɛɥɢɰɵ 7 u 10, 10 u 10 ɢ ɬ.ɞ. ɫ 50 - 80 ɢ ɞɚɠɟ ɛɨɥɶɲɢɦ ɱɢɫɥɨɦ ɫɬɟɩɟɧɟɣ ɫɜɨɛɨɞɵ). ɉɪɢɜɟɞɟɧɧɵɟ ɧɢɠɟ ɬɚɛɥɢɰɵ ɫɨɫɬɚɜɥɟɧɵ, ɤɚɤ ɩɪɚɜɢɥɨ, ɞɥɹ ɱɢɫɥɚ ɫɬɟɩɟɧɟɣ ɫɜɨɛɨɞɵ ɢɥɢ ɨɛɴɟɦɚ ɜɵɛɨɪɤɢ ɨɬ 1 ɞɨ 1000 ɢ ɞɥɹ ɭɪɨɜɧɟɣ ɡɧɚɱɢɦɨɫɬɢ 5%, 1% ɢ 0,1%. Ⱦɭɦɚɟɬɫɹ, ɱɬɨ ɬɚɤɚɹ ɫɬɚɧɞɚɪɬɢɡɚɰɢɹ ɨɛɥɟɝɱɢɬ ɩɨɥɶɡɨɜɚɧɢɟ ɬɚɛɥɢɰɚɦɢ.
ɂɫɯɨɞɹ ɢɡ ɜɵɫɤɚɡɚɧɧɵɯ ɜ ɝɥ. V ɫɨɨɛɪɚɠɟɧɢɣ ɨ ɛɨɥɶɲɟɣ ɨɩɚɫɧɨɫɬɢ ɨɲɢɛɨɤ I ɪɨɞɚ, ɱɟɦ
II ɪɨɞɚ, ɜɫɟ ɬɚɛɥɢɰɵ ɩɪɢɜɨɞɹɬɫɹ ɞɥɹ ɞɜɭɫɬɨɪɨɧɧɟɝɨ ɤɪɢɬɟɪɢɹ (ɱɬɨ ɭɦɟɧɶɲɚɟɬ ɜɟɪɨɹɬɧɨɫɬɶ ɨɲɢɛɨɤ I ɪɨɞɚ ɢ ɭɜɟɥɢɱɢɜɚɟɬ ɜɟɪɨɹɬɧɨɫɬɶ ɨɲɢɛɨɤ II ɪɨɞɚ). ɂɡ ɷɬɢɯ ɠɟ ɫɨɨɛɪɚɠɟɧɢɣ ɪɟɤɨɦɟɧɞɭɟɦ ɱɢɬɚɬɟɥɸ ɜ ɫɥɭɱɚɟ, ɟɫɥɢ ɜ ɬɚɛɥɢɰɟ ɧɟ ɩɪɢɜɟɞɟɧɨ ɤɪɢɬɢɱɟɫɤɨɟ ɡɧɚɱɟɧɢɟ ɞɥɹ ɩɨɥɭɱɟɧɧɨɣ ɢɦ ɫɬɚɬɢɫɬɢɱɟɫɤɨɣ ɯɚɪɚɤɬɟɪɢɫɬɢɤɢ, ɛɪɚɬɶ ɫ ɧɟɤɨɬɨɪɵɦ ɡɚɩɚɫɨɦ ɛɥɢɠɚɣɲɟɟ ɛɨɥɶɲɟɟ ɤɪɢɬɢɱɟɫɤɨɟ ɡɧɚɱɟɧɢɟ (ɢɥɢ ɠɟ ɩɪɢɛɟɝɧɭɬɶ ɤ ɢɧɬɟɪɩɨɥɹɰɢɢ).
Ɉɬɦɟɬɢɦ ɬɚɤɠɟ, ɱɬɨ ɩɪɢ ɪɚɛɨɬɟ ɧɚɞ ɬɚɛɥɢɰɚɦɢ ɛɵɥɢ ɨɛɧɚɪɭɠɟɧɵ ɨɩɟɱɚɬɤɢ ɜ ɧɟɤɨɬɨɪɵɯ ɢɡɞɚɧɢɹɯ7.
ɌɚɛɥɢɰɚȺ
ɇɨɪɦɚɥɶɧɨɟ ɪɚɫɩɪɟɞɟɥɟɧɢɟ8
Ɋɢɫ. 31. Ⱦɨɥɢ ɩɥɨɳɚɞɢ (Ɋ) ɩɨɞ ɧɨɪɦɚɥɶɧɨɣ ɤɪɢɜɨɣ ɜ ɩɪɟɞɟɥɚɯ ɨɬ zɤɪ ɞɨ zɤɪ
zɤɪ |
|
Ɋ |
zɤɪ |
Ɋ |
zɤɪ |
Ɋ |
zɤɪ |
Ɋ |
0,01 |
|
0,00000 |
02 |
01596 |
04 |
03191 |
06 |
04784 |
01 |
|
00798 |
03 |
02393 |
05 |
03988 |
07 |
05581 |
|
[255] |
|
|
|
|
|
|
|
|
ɉɪɨɞɨɥɠɟɧɢɟ ɬɚɛɥ. Ⱥ |
|
|
|
|
|
||
zɤɪ |
|
Ɋ |
zɤɪ |
Ɋ |
zɤɪ |
Ɋ |
zɤɪ |
Ɋ |
|
|
|
|
|
|
|
|
|
08 |
|
06376 |
57 |
43132 |
06 |
71086 |
1,55 |
0,87886 |
09 |
|
07171 |
58 |
43809 |
07 |
71538 |
56 |
88124 |
0,10 |
|
0,07966 |
59 |
44481 |
08 |
71986 |
57 |
88358 |
11 |
|
08759 |
0,60 |
0,45149 |
09 |
72429 |
58 |
88589 |
12 |
|
09552 |
61 |
45814 |
3,10 |
0,72867 |
59 |
88817 |
13 |
|
10348 |
62 |
46474 |
11 |
73300 |
1,60 |
0,89040 |
6 Ɍɚɦ ɠɟ, ɬɚɛɥ. Ȼ ɧɚ ɫ. 300, 301.
7 ɋɬɚɬɢɫɬɢɱɟɫɤɢɟ ɦɟɬɨɞɵ..., ɫ. 300, ɩɨɫɥɟɞɧɹɹ ɫɬɪɨɤɚ: ɜɦɟɫɬɨ 7,251 ɞɨɥɠɧɨ ɛɵɬɶ 6,25l, ɚ ɜɦɟɫɬɨ 6,815 ɞɨɥɠɧɨ ɛɵɬɶ 7,815; ȼɟɧɟɰɤɢɣ ɂ. Ƚ., ȼɟɧɟɰɤɚɹ ȼ. ɂ. Ɉɫɧɨɜɧɵɟ ɦɚɬɟɦɚɬɢɤɨ-ɫɬɚɬɢɫɬɢɱɟɫɤɢɟ ɩɨɧɹɬɢɹ ɢ ɮɨɪɦɭɥɵ ɜ ɷɤɨɧɨɦɢɱɟɫɤɨɦ ɚɧɚɥɢɡɟ. M., 1979, ɫ. 413, ɩɨɫɥɟɞɧɹɹ ɫɬɪɨɤɚ: ɜɦɟɫɬɨ 1,77 ɞɨɥɠɧɨ ɛɵɬɶ 2,77.
8 ȼɟɧɟɰɤɢɣ ɂ. Ƚ., ȼɟɧɟɰɤɚɹ ȼ. ɂ. Ɉɫɧɨɜɧɵɟ ɦɚɬɟɦɚɬɢɤɨ-ɫɬɚɬɢɫɬɢɱɟɫɤɢɟ ɩɨɧɹɬɢɹ..., ɫ, 402 - 404.
181
14 |
|
11134 |
63 |
47131 |
12 |
73729 |
61 |
89260 |
15 |
|
11924 |
64 |
47783 |
13 |
74152 |
62 |
89477 |
16 |
|
12712 |
65 |
48431 |
14 |
74571 |
63 |
89690 |
17 |
|
13499 |
66 |
49075 |
15 |
74986 |
64 |
89899 |
18 |
|
14285 |
67 |
49714 |
16 |
75395 |
65 |
90106 |
19 |
|
15069 |
68 |
50350 |
17 |
75800 |
66 |
90309 |
0,20 |
|
0,15852 |
69 |
50981 |
18 |
76200 |
67 |
90508 |
21 |
|
16633 |
0,70 |
0,51607 |
19 |
76595 |
68 |
90704 |
22 |
|
17413 |
71 |
52230 |
3,20 |
0,76986 |
69 |
90897 |
23 |
|
18191 |
72 |
52848 |
21 |
77372 |
1,70 |
0,91087 |
24 |
|
18967 |
73 |
53461 |
22 |
77754 |
71 |
91273 |
25 |
|
19741 |
74 |
54070 |
23 |
78130 |
72 |
91457 |
26 |
|
20514 |
75 |
54675 |
24 |
78502 |
73 |
91637 |
27 |
|
21284 |
76 |
55275 |
25 |
78870 |
74 |
9144 |
28 |
|
22052 |
77 |
55870 |
26 |
79233 |
75 |
91588 |
29 |
|
22818 |
78 |
56461 |
27 |
79592 |
76 |
92159 |
0,30 |
|
0,23582 |
79 |
57047 |
28 |
79945 |
77 |
92327 |
31 |
|
24344 |
0,80 |
0,57629 |
29 |
80295 |
78 |
92492 |
32 |
|
25103 |
81 |
58206 |
3,30 |
0,80640 |
79 |
92655 |
33 |
|
25860 |
82 |
58778 |
31 |
80980 |
1,80 |
0,92814 |
34 |
|
26614 |
83 |
59346 |
32 |
81316 |
81 |
92970 |
35 |
|
27366 |
84 |
59909 |
33 |
81648 |
82 |
93124 |
36 |
|
28115 |
85 |
60468 |
34 |
81975 |
83 |
93275 |
37 |
|
28862 |
86 |
61021 |
35 |
82298 |
84 |
93423 |
38 |
|
29605 |
87 |
61570 |
36 |
82617 |
'85 |
93569 |
39 |
|
30346 |
88 |
62114 |
37 |
82931 |
86 |
93711 |
0,40 |
|
0,31084 |
89 |
62653 |
38 |
83241 |
87 |
93852 |
41 |
|
31819 |
0,90 |
0,63188 |
39 |
83547 |
88 |
93989 |
42 |
|
32552 |
91 |
63718 |
3,40 |
0,83849 |
89 |
44 124 |
43 |
|
33280 |
92 |
64243 |
41 |
84146 |
1,90 |
0,94257 |
44 |
|
34006 |
93 |
64763 |
42 |
84439 |
91 |
94387 |
45 |
|
34729 |
94 |
65278 |
43 |
84728 |
92 |
94514 |
46 |
|
35448 |
95 |
65789 |
44 |
85013 |
93 |
94639 |
47 |
|
36164 |
96 |
66294 |
45 |
85294 |
94 |
94762 |
48 |
|
36877 |
97 |
66795 |
46 |
85571 |
95 |
94882 |
49 |
|
37587 |
98 |
67291 |
47 |
85844 |
96 |
95000 |
0,50 |
|
0,38292 |
99 |
67783 |
48 |
86113 |
97 |
95116 |
51 |
|
38995 |
1,00 |
0,68269 |
49 |
86378 |
98 |
95230 |
52 |
|
39694 |
01 |
68750 |
3,505 |
0,86639 |
99 |
95341 |
53 |
|
40389 |
02 |
69227 |
51 |
86696 |
2,00 |
0,95450 |
54 |
|
41080 |
03 |
69699 |
52 |
87149 |
01 |
95557 |
55 |
|
41768 |
04 |
70166 |
53 |
87398 |
02 |
95662 |
56 |
|
42452 |
05 |
70628 |
54 |
87644 |
03 |
95764 |
|
[256] |
|
|
|
|
|
|
|
|
ɉɪɨɞɨɥɠɟɧɢɟ ɬɚɛɥ. Ⱥ. |
|
|
|
|
|
||
zɤɪ |
|
|
zɤɪ |
|
zɤɪ |
|
zɤɪ |
|
|
|
Ɋ |
Ɋ |
Ɋ |
Ɋ |
|||
|
|
|
|
|
|
|
|
|
04 |
|
95865 |
53 |
98859 |
02 |
99747 |
51 |
99955 |
05 |
|
95964 |
54 |
98891 |
03 |
99755 |
52 |
99957 |
06 |
|
96060 |
55 |
98923 |
04 |
99763 |
53 |
99958 |
07 |
|
96155 |
56 |
98953 |
05 |
999771 |
54 |
99960 |
182
08 |
96247 |
57 |
98983 |
06 |
99779 |
55 |
99961 |
09 |
96338 |
58 |
99012 |
07 |
99786 |
56 |
99963 |
2,10 |
0,96427 |
59 |
99040 |
08 |
99793 |
57 |
99964 |
11 |
96514 |
2,60 |
0,99068 |
09 |
99800 |
58 |
99966 |
12 |
96599 |
61 |
99095 |
3,10 |
0,99806 |
59 |
99967 |
13 |
96683 |
62 |
99121 |
11 |
99813 |
3,60 |
0,99968 |
14 |
96765 |
63 |
99146 |
12 |
99819 |
61 |
99969 |
15 |
96844 |
64 |
99171 |
13 |
99825 |
62 |
99971 |
16 |
96923 |
65 |
99195 |
14 |
99831 |
63 |
99972 |
17 |
96999 |
66 |
99219 |
15 |
99837 |
64 |
99973 |
18 |
97074 |
67 |
99241 |
16 |
99842 |
3,65 |
0,99974 |
19 |
97148 |
68 |
99263 |
17 |
99848 |
66 |
99975 |
2,20 |
0,97219 |
69 |
99285 |
18 |
99853 |
67 |
99976 |
21 |
97289 |
2,70 |
0,99307 |
19 |
99858 |
68 |
99977 |
22 |
97358 |
71 |
99327 |
3,20 |
0,99863 |
69 |
99978 |
23 |
97425 |
72 |
99347 |
21 |
99867 |
3,70 |
0,99978 |
24 |
97491 |
73 |
99367 |
22 |
99872 |
71 |
99979 |
25 |
0,97555 |
74 |
99386 |
23 |
99876 |
72 |
99980 |
26 |
97618 |
75 |
99404 |
24 |
99880 |
73 |
99981 |
27 |
97679 |
76 |
99422 |
25 |
99855 |
74 |
99982 |
28 |
97739 |
77 |
99439 |
26 |
99889 |
75 |
99982 |
29 |
97789 |
78 |
99456 |
27 |
99892 |
76 |
99983 |
2,30 |
0,97855 |
79 |
99473 |
28 |
99896 |
77 |
99984 |
31 |
97911 |
2,80 |
0,99489 |
29 |
99900 |
78 |
99984 |
32 |
97966 |
81 |
99505 |
3,30 |
0,99903 |
79 |
99985 |
33 |
98019 |
82 |
99520 |
31 |
99907 |
3,80 |
0,99986 |
34 |
98072 |
83 |
99535 |
32 |
99910 |
81 |
99986 |
35 |
98123 |
84 |
99549 |
33 |
99913 |
82 |
99987 |
36 |
98172 |
85 |
99563 |
34 |
99916 |
83 |
9987 |
37 |
98221 |
86 |
99576 |
35 |
99919 |
84 |
99988 |
38 |
98269 |
87 |
99590 |
36 |
99922 |
85 |
99988 |
39 |
98315 |
88 |
99602 |
37 |
99925 |
86 |
99989 |
2,40 |
0,98360 |
89 |
99615 |
38 |
99928 |
87 |
99989 |
41 |
98405 |
2,90 |
0,99627 |
39 |
99930 |
88 |
99990 |
42 |
98448 |
91 |
99639 |
3,40 |
0,99933 |
89 |
99990 |
43 |
98490 |
92 |
99650 |
41 |
99935 |
3,90 |
0,99990 |
44 |
98531 |
93 |
99661 |
42 |
99937 |
91 |
99991 |
45 |
98571 |
94 |
99672 |
43 |
99940 |
92 |
99991 |
46 |
98611 |
2,95 |
0,99682 |
44 |
99942 |
93 |
99992 |
47 |
98649 |
96 |
99692 |
45 |
99944 |
94 |
99992 |
48 |
98686 |
97 |
99702 |
46 |
99946 |
95 |
99992 |
49 |
98723 |
98 |
99712 |
47 |
99948 |
96 |
99992 |
2,50 |
0,98758 |
99 |
99721 |
48 |
99950 |
97 |
99993 |
51 |
98793 |
3,00 |
0,99730 |
49 |
99952 |
98 |
99993 |
52 |
98826 |
01 |
99739 |
3,50 |
0,99953 |
99 |
99993 |
[257]
Ɍɚɛɥɢɰɚ Ȼ.
183
Ɂɧɚɱɟɧɢɟ F2ɨ ɜ ɡɚɜɢɫɢɦɨɫɬɢ ɨɬ ɱɢɫɥɚ ɫɬɟɩɟɧɟɣ ɫɜɨɛɨɞɵ (f) ɢ ɭɪɨɜɧɹ ɡɧɚɱɢɦɨɫɬɢ9
|
ɍɪɨɜɟɧɶ ɡɧɚɱɢɦɨɫɬɢ. |
|
ɍɪɨɜɟɧɶ ɡɧɚɱɢɦɨɫɬɢ, |
|
ɍɪɨɜɟɧɶ ɡɧɚɱɢɦɨɫɬɢ, % |
||||||
f |
% |
|
|
f |
% |
|
|
f |
|||
|
|
|
|
|
|
|
|||||
|
|
|
|
|
|
|
|
|
|||
|
5 |
1 |
0,1 |
|
5 |
1 |
0,1 |
|
5 |
1 |
0,1 |
1 |
3,84 |
6,63 |
10,83 |
16 |
26,30 |
32,00 |
39,25 |
40 |
55,76 |
63,69 |
73,40 |
2 |
5,99 |
9,21 |
13,81 |
17 |
27,59 |
33,41 |
40,79 |
50 |
67,50 |
76,15 |
86,66 |
3 |
7,81 |
11,34 |
16,27 |
18 |
28,87 |
34,80 |
42,31 |
60 |
79,08 |
8838 |
99,61 |
4 |
9,49 |
13,28 |
18,46 |
19 |
30,14 |
36,19 |
43,82 |
70 |
90,53 |
100,42 |
112,33 |
5 |
11,07 |
15,09 |
20,52 |
20 |
31,41 |
37,57 |
45,31 |
80 |
101,88 |
112,33 |
124,84 |
6 |
12,59 |
16,81 |
22,46 |
21 |
32,67 |
38,93 |
46,80 |
90 |
113,15 |
124,12 |
137,21 |
7 |
14,07 |
18,47 |
24,32 |
22 |
33,92 |
40,29 |
48,27 |
100 |
124,34 |
135,81 |
149,45 |
8 |
15,51 |
20,09 |
26,12 |
23 |
35,17 |
41,63 |
49,73 |
200 |
233,99 |
249,44 |
267,50 |
9 |
16,92 |
21,67 |
27,88 |
24 |
36,41 |
42,98 |
51,18 |
300 |
341,40 |
359,91 |
— |
10 |
18,31 |
23,21 |
29,59 |
25 |
37,65 |
44.31 |
52,62 |
400 |
447,63 |
468,72 |
— |
11 |
19,67 |
24,72 |
31,26 |
26 |
38,88 |
45,64 |
54,05 |
500 |
553,13 |
576,49 |
— |
12 |
21,03 |
26,22 |
32,91 |
27 |
40,11 |
46,96 |
55,48 |
600 |
658,09 |
683,52 |
— |
13 |
22,37 |
27,69 |
34,53 |
28 |
41,34 |
48,28 |
56,89 |
700 |
762,66 |
789,97 |
— |
14 |
23,68 |
29,14 |
36,12 |
29 |
42,56 |
49,59 |
58,30 |
800 |
866,91 |
895,98 |
— |
15 |
25,00 |
30,58 |
37,70 |
30 |
43,77 |
50,89 |
59,70 |
1000 |
1074,68 |
1106,97 |
— |
|
[258] |
|
|
|
|
|
|
|
|
|
|
Ɍɚɛɥɢɰɚ ȼ.
Ʉɪɢɬɢɱɟɫɤɢɟ ɡɧɚɱɟɧɢɹ ɤɨɷɮɮɢɰɢɟɧɬɚ ɪɚɧɝɨɜɨɣ ɤɨɪɪɟɥɹɰɢɢ U
|
Ɉɛɴɟɦ |
ɍɪɨɜɟɧɶ ɡɧɚɱɢɦɨɫɬɢ, % |
Ɉɛɴɟɦ |
ɍɪɨɜɟɧɶ ɡɧɚɱɢɦɨɫɬɢ, % |
||||
|
|
|
|
|
|
|
|
|
|
ɜɵɛɨɪɤɢ |
5 |
1 |
0,1 |
ɜɵɛɨɪɤɢ |
5 |
1 |
0,1 |
|
|
|
|
|
|
|
|
|
6 |
0,829 |
1,000 |
_ |
25 |
0,398 |
0,510 |
0,618 |
|
7 |
0,745 |
0,893 |
1,000 |
30 |
0,362 |
0,466 |
0,570 |
|
8 |
0,691 |
0,857 |
0,952 |
35 |
0,333 |
0,429 |
0,534 |
|
9 |
0,683 |
0,817 |
0,917 |
40 |
0,311 |
0,402 |
0,501 |
|
10 |
0,636 |
0,782 |
0,891 |
45 |
0,294 |
0,380 |
0,475 |
|
11 |
0,618 |
0,754 |
0,867 |
50 |
0,279 |
0,361 |
0,450 |
|
12 |
0,580 |
0,727 |
0,823 |
60 |
0,254 |
0,330 |
0,415 |
|
13 |
0,555 |
0,698 |
0,801 |
70 |
0,235 |
0,306 |
0,385 |
|
14 |
0,534 |
0,675 |
0,793 |
80 |
0,220 |
0,286 |
0,361 |
|
15 |
0,518 |
0,654 |
0,760 |
90 |
0,207 |
0,270 |
0,341 |
|
16 |
0,500 |
0,632 |
0,741 |
100 |
0,196 |
0,257 |
0,324 |
|
17 |
0,485 |
0,615 |
0,734 |
150 |
0,160 |
0,209 |
0,265 |
|
18 |
0,472 |
0,598 |
0,709 |
200 |
0,139 |
0,182 |
0,231 |
|
19 |
0,458 |
0,583 |
0,694 |
500 |
0,087 |
0,115 |
0,148 |
|
20 |
0,445 |
0,568 |
0,679 |
1000 |
0,062 |
0,081 |
0,104 |
|
Ʉɪɢɬɢɱɟɫɤɢɟ ɡɧɚɱɟɧɢɹ ɩɪɢ n < 12 ɪɚɫɫɱɢɬɚɧɵ ɧɚɦɢ ɩɨ ɬɚɛɥɢɰɟ ɬɨɱɧɨɝɨ ɪɚɫɩɪɟɞɟɥɟɧɢɹ S (Ʉɟɧɞɟɥ Ɇ. Ɋɚɧɝɨɜɵɟ ɤɨɪɪɟɥɹɰɢɢ. Ɇ., 1975, ɫ. 188, 189). Ɂɧɚɱɟɧɢɹ ɩɪɢ 12<n<30 ɩɪɢ ɭɪɨɜɧɟ ɡɧɚɱɢɦɨɫɬɢ 5% ɢ 1% ɡɚɢɦɫɬɜɨɜɚɧɵ ɢɡ ɤɧɢɝɢ: Ɂɚɤɫ Ʌ. ɋɬɚɬɢɫɬɢɱɟɫɤɨɟ ɨɰɟɧɢɜɚɧɢɟ. Ɇ., 1976, ɫ. 369, ɨɫɬɚɥɶɧɵɟ ɡɧɚɱɟɧɢɹ ɪɚɫɫɱɢɬɚɧɵ ɩɨ ɮɨɪɦɭɥɟ, ɩɨɥɭɱɟɧɧɨɣ ɢɡ (V,8,4).
9 ɑɚɫɬɶ ɬɚɛɥɢɰɵ ɡɚɢɦɫɬɜɨɜɚɧɚ ɢɡ ɤɧ.: Ɂɚɤɫ Ʌ. ɋɬɚɬɢɫɬɢɱɟɫɤɨɟ ɨɰɟɧɢɜɚɧɢɟ. Ɇ., 1976, ɫ. 132—134, ɱɚɫɬɶ — ɢɡ ɤɧ. Ɉɭɷɧ Ⱦ. Ȼ. ɋɛɨɪɧɢɤ ɫɬɚɬɢɫɬɢɱɟɫɤɢɯ ɬɚɛɥɢɰ. Ɇ., 1966, ɫ. 49—55.
184