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ɋɪɚɜɧɢɦ ɷɬɨɬ ɪɟɡɭɥɶɬɚɬ ɫ ɜɤɥɚɞɨɦ ɜ ɜɨɫɩɪɢɢɦɱɢɜɨɫɬɶ ܳ
ɦɚɝɧɢɬɧɨɝɨ ɬɨɤɚ, ɞɥɹ ɷɬɨɝɨ ɩɪɨɢɧɬɟɝɪɢɪɭɟɦ ɩɨ x
ሺxሻ
݁ଶ݊
ɜ ɪɟɡɭɥɶɬɚɬɟ ɩɨɥɭɱɢɦ െߜ
ఈఉ
Ȁ݉. Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, ɷɬɢ ɜɤɥɚɞɵ ɫɨɤɪɚɳɚɸɬ
ɩɨɫɥɟɞɧɟɟ ɫɥɚɝɚɟɦɨɟ ɜ (4.16),
ଵ
ఈఉ
ሺ
ߗǡxǡx
ሻ
ɨɬ ɞɢɚ-
ଵ
ɞɪɭɝ ɞɪɭɝɚ. Ʉɚɤ ɦɵ ɜɢɞɢɦ, ɫɨɤɪɚɳɟɧɢɟ ɷɬɢɯ ɜɤɥɚɞɨɜ ɩɪɢ ߗ՜Ͳ ɨɛɟɫɩɟɱɢɜɚɟɬ ɤɨ-
ɧɟɱɧɨɫɬɶ ɩɪɨɜɨɞɢɦɨɫɬɢ ɫɢɫɬɟɦɵ.
ɉɨɫɥɟ ɭɫɪɟɞɧɟɧɢɹ ɩɨ ɛɟɫɩɨɪɹɞɤɭ, ɮɭɧɤɰɢɹ
ሺ
ሻ
௦௧௧
න݀
ௗ
x
ଵ
ሺ
ఈఉ
ߗ՜Ͳǡxǡx
ߎ
ሻሺ
ଵ
ͶǤͷ
ሻ
ɫɬɚɧɨɜɢɬɫɹ ɬɪɚɧɫɥɹɰɢɨɧɧɨ-ɢɧɜɚɪɢɚɧɬɧɨɣ ɢ ɡɚɜɢɫɹɳɟɣ ɬɨɥɶɤɨ ɨɬ ɪɚɡɧɨɫɬɢ x െx
ɬɨɝɞɚ ɜ ɪɚɜɟɧɫɬɜɟ (4.55) ɫɥɟɜɚ ɫɬɨɢɬ ɮɭɪɶɟ-ɨɛɪɚɡ ɩɨ ɩɪɨɫɬɪɚɧɫɬɜɟɧɧɵɦ ɤɨɨɪɞɢ-
ɧɚɬɚɦ ɩɪɢ ɜɨɥɧɨɜɨɦ ɜɟɤɬɨɪɟ k ൌͲ, ɬɚɤɢɦ ɨɛɪɚɡɨɦ, ɜ ɷɬɨɦ ɫɥɭɱɚɟ ɢɦɟɟɦ
ଶ
ఈఉ
݁
݉
݊ǡ
ሺ
ͶǤͷ
ɝɞɟ ݊
ሺ
ሻ
௦௧௧
ሺ
ߎ
ሺxሻ
ൌ݊
ɭɫɪɟɞɧɟɧɧɚɹ ɩɨ ɛɟɫɩɨɪɹɞɤɭ ɤɨɧɰɟɧɬɪɚɰɢɹ ɷɥɟɤɬɪɨɧɨɜ.
ߗ՜Ͳǡk ൌͲሻൌߜ
ఈఉ
Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, ɦɵ ɩɨɤɚɡɚɥɢ, ɱɬɨ ɩɪɢ ߗ՜Ͳ ɜ ɞɢɧɚɦɢɱɟɫɤɭɸ ɜɨɫɩɪɢɢɦɱɢ-
ɜɨɫɬɶ ɜɧɨɫɢɬ ɜɤɥɚɞ ɬɨɥɶɤɨ (4.44), ɬɨ ɟɫɬɶ ɦɨɠɧɨ ɧɚɩɢɫɚɬɶ
ሺ
ሻ
ሺ
ܳ
ߗǡxǡx
ൌെߗ
ଶ
݁
ͺߨ݉
Ԝൈܩ
න݀
ଶ
݅
ఈఉ
ஶ
ିஶ
݀݊
߱Ԝቆ
ோ
ሺ
߱ߗǡyǡx
ଵ
ி
݀߱
ȼɜɟɞɟɦ ɬɟɩɟɪɶ ɩɪɨɜɨɞɢɦɨɫɬɶ ɫɢɫɬɟɦɵ ߪ
ሺ
ߗǡxሻൌන݀
݆
ఈ
ௗ
ሻ
ൌߎ
ሺ߱ሻ
x
ଵߪఈఉ
ఈఉ
ቇ൫
ଵ
ሺ
ሺ
ఈ
െ
x
ሻ
ܩ
൫߱ǡyଵǡx൯Ǥ
ሺ
ఈఉ
ߗǡxǡx
ߗǡxǡx
ఈ
൯ቀ
y
ߗǡxǡx
ሻ
ଵ
ሻ
ൌ
ଵ
ఉ
ఉ
െ
భ
ଵ
ǡߗሻǡ
y
ቁൈ
భ
ሺ
ͶǤͷͺ
ሺ
ͶǤͷͻ
x
ሻ
ଵ
ሺ
ܧ
x
ఉ
ɢɫɩɨɥɶɡɭɹ ɫɨɨɬɧɨɲɟɧɢɟ
ሺ
݆
ఈ
xǡߗሻൌ
ͳ
න݀xଵܳ
ܿ
ఈఉ
ሺ
ߗǡxǡx
ሻ
ሺ
ܣ
x
ଵ
ǡߗሻǡ
ఉ
ଵ
ሺ
ͶǤͲ
ɚ ɬɚɤɠɟ ɫɜɹɡɶ ɧɚɩɪɹɠɟɧɧɨɫɬɢ ɩɨɥɹ ɢ ɜɟɤɬɨɪɧɨɝɨ ɩɨɬɟɧɰɢɚɥɚ
ܧ
ሺ
ఉ
x
ǡߗሻൌ݅
ଵ
ߗ
ሺ
x
ܣ
ǡߗሻǡ
ఉ
ଵ
ܿ
ሺ
ͶǤͳ
ɩɨɥɭɱɢɦ ɮɨɪɦɭɥɭ ɞɥɹ ɩɪɨɜɨɞɢɦɨɫɬɢ ɫɢɫɬɟɦɵ
ஶ
ଶ
݁
ሺ
ߪ
ఈఉ
ߗǡxǡx
ሻ
ଵ
ൌ
ͺߨ݉
ൈܩ
න݀
ଶ
ିஶ
ோ
ሺ
߱ߗǡyǡx
߱Ԝቆ
݀݊
ி
݀߱
ሻ
ଵ
ሺ߱ሻ
ܩ
ఈ
ቇ൫
x
൫߱ǡyଵǡx൯ǡ
െ
ఈ
y
൯ቀ
ఉ
ఉ
െ
x
ቁൈԜ
y
భ
భ
ሺ
ͶǤʹ
,
ଵ
ሻ
ሻ
ሻ
ሻ
ሻ
ሻ
ɝɞɟ ɩɨɫɥɟ ɞɟɣɫɬɜɢɹ ɨɩɟɪɚɬɨɪɨɜ ɧɭɠɧɨ ɩɨɥɨɠɢɬɶ y ൌx ɢ y
41
ൌx
.
ଵ
ଵ
ȼ ɞɚɥɶɧɟɣɲɟɦ ɧɚɫ ɛɭɞɟɬ ɢɧɬɟɪɟɫɨɜɚɬɶ ɩɪɨɜɨɞɢɦɨɫɬɶ ɫɢɫɬɟɦɵ ɜ ɨɞɧɨɪɨɞɧɨɦ
ɩɨɥɟ ܧ
ሺ
ఉ
x
ǡߗሻؠܧ
ଵ
ሺߗሻ
, ɢ ɤɪɨɦɟ ɬɨɝɨ ɦɵ ɛɭɞɟɦ ɪɚɫɫɦɚɬɪɢɜɚɬɶ ɥɢɛɨ ɨɞɧɨɦɟɪ-
ఉ
ɧɭɸ, ɥɢɛɨ ɢɡɨɬɪɨɩɧɭɸ ɫɢɫɬɟɦɭ ɷɥɟɤɬɪɨɧɨɜ, ɬɚɤ ɱɬɨ ɬɟɧɡɨɪɧɚɹ ɫɬɪɭɤɬɭɪɚ ɩɪɨɜɨɞɢɦɨɫɬɢ ɛɭɞɟɬ ɬɪɢɜɢɚɥɶɧɨɣ, ɜ ɷɬɨɦ ɫɥɭɱɚɟ ɮɨɪɦɭɥɭ (4.59) ɦɨɠɧɨ ɩɟɪɟɩɢɫɚɬɶ ɬɚɤ
ɝɞɟ ɞɥɹ ߪሺߗǡx
ߪሺߗǡxሻൌ
ሻ
ɢɦɟɟɦ
ଶ
݁
ͺߨ݉
ሺ
݆
ߗǡxሻൌߪሺߗǡxሻܧ
ఈ
Ԝන݀
ଶ
݀
ௗ
x
ଵ
ஶ
Ԝන ݀
ିஶ
߱Ԝቆ
݀݊
ி
݀߱
ሺ߱ሻ
ఈ
ሺߗሻ
ቇ൫
ሺ
ǡ
ఈ
ఈ
ఈ
x
െ
y
൯ቀ
x
భ
െ
ఈ
y
భ
ቁൈ
ͶǤ͵
ሻ
Ԝൈܩ
ோ
ሺ
߱ߗǡyǡx
ሻ
ܩ
ଵ
൫߱ǡyଵǡx൯ǡ
ሺ
ͶǤͶ
ሻ
ɝɞɟ ݀ ɪɚɡɦɟɪɧɨɫɬɶ ɩɪɨɫɬɪɚɧɫɬɜɚ, ɚ ɩɨ ɢɧɞɟɤɫɭ ߙ ɩɨɞɪɚɡɭɦɟɜɚɟɬɫɹ ɫɭɦɦɢɪɨɜɚɧɢɟ. ɏɨɬɹ ɜɟɥɢɱɢɧɚ ߪሺߗǡx
ሻ
ɮɨɪɦɚɥɶɧɨ ɡɚɜɢɫɢɬ ɨɬ ɬɨɱɤɢ ɩɪɨɫɬɪɚɧɫɬɜɚ x, ɩɨɫɥɟ
ɭɫɪɟɞɧɟɧɢɹ ɩɨ ɛɟɫɩɨɪɹɞɤɭ ɷɬɚ ɡɚɜɢɫɢɦɨɫɬɶ ɢɫɱɟɡɧɟɬ ɢ ɷɬɚ ɮɭɧɤɰɢɹ ɛɭɞɟɬ ɨɩɢɫɵ-
ɜɚɬɶ ɩɪɨɜɨɞɢɦɨɫɬɶ ɫɢɫɬɟɦɵ ߪሺߗǡkሻ ɩɪɢ ɜɨɥɧɨɜɨɦ ɜɟɤɬɨɪɟ k ൌͲ ɢ ɦɵ
ɲɟɦ ɛɭɞɟɦ ɨɛɨɡɧɚɱɚɬɶ ɟɟ ߪሺߗ
ሻ
.
ɜ ɞɚɥɶɧɟɣ-
Ʌɢɬɟɪɚɬɭɪɚ
ȼɵɪɚɠɟɧɢɟ ɞɥɹ ɦɚɰɭɛɚɪɨɜɫɤɨɝɨ ɨɬɤɥɢɤɚ ɧɚ ɩɟɪɟɦɟɧɧɨɟ ɷɥɟɤɬɪɨɦɚɝɧɢɬɧɨɟ
ɩɨɥɟ ɩɨɥɭɱɟɧɨ ɜ [4], ɝɥ. XI. Ɍɟɯɧɢɤɚ ɚɧɚɥɢɬɢɱɟɫɤɨɝɨ ɩɪɨɞɨɥɠɟɧɢɹ ɫ ɞɢɫɤɪɟɬɧɨɝɨ
ɪɹɞɚ ɦɧɢɦɵɯ ɬɨɱɟɤ ɧɚ ɜɫɸ ɜɟɪɯɧɸɸ ɩɨɥɭɩɥɨɫɤɨɫɬɶ ɨɩɢɫɚɧɚ [4], ɝɥ. XI ɢ ɜ [5],
§ 37. ȼɵɜɨɞ ɬɨɠɞɟɫɬɜ ɞɥɹ ɩɪɨɢɡɜɨɞɧɵɯ ɨɬ ɮɭɧɤɰɢɣ Ƚɪɢɧɚ ɞɚɧ ɜ [3], § 19.
Ʉɨɧɬɪɨɥɶɧɵɟ ɜɨɩɪɨɫɵ
1. ȼɵɩɢɫɚɬɶ ɝɪɚɞɢɟɧɬɧɵɣ ɜɤɥɚɞ ɜ ɨɩɟɪɚɬɨɪ ɩɥɨɬɧɨɫɬɢ ɬɨɤɚ ɫɢɫɬɟɦɵ ɷɥɟɤ-
ɬɪɨɧɨɜ.
2. ȼɵɩɢɫɚɬɶ ɞɢɚɦɚɝɧɢɬɧɵɣ ɜɤɥɚɞ ɜ ɨɩɟɪɚɬɨɪ ɩɥɨɬɧɨɫɬɢ ɬɨɤɚ ɫɢɫɬɟɦɵ ɷɥɟɤ-
ɬɪɨɧɨɜ.
3. ɑɬɨ ɬɚɤɨɟ ɤɚɥɢɛɪɨɜɨɱɧɵɟ ɩɪɟɨɛɪɚɡɨɜɚɧɢɹ ɷɥɟɤɬɪɨɦɚɝɧɢɬɧɨɝɨ ɩɨɥɹ?
4. Ʉɚɤ ɩɪɟɨɛɪɚɡɭɸɬɫɹ ߰-ɨɩɟɪɚɬɨɪɵ ɩɪɢ ɤɚɥɢɛɪɨɜɨɱɧɵɯ ɩɪɟɨɛɪɚɡɨɜɚɧɢɹɯ?
5. Ʉ ɤɚɤɨɦɭ ɬɨɠɞɟɫɬɜɭ ɞɥɹ ɮɭɧɤɰɢɣ Ƚɪɢɧɚ ɩɪɢɜɨɞɢɬ ɬɪɟɛɨɜɚɧɢɟ ɤɚɥɢɛɪɨɜɨɱ-
ɧɨɣ ɢɧɜɚɪɢɚɧɬɧɨɫɬɢ?
42
Ƚɥɚɜɚ 5
ȼɕɊɈɀȾȿɇɇɕɃ ɗɅȿɄɌɊɈɇɇɕɃ ȽȺɁ
ȼ ɉɈɌȿɇɐɂȺɅȿ ɉɊɂɆȿɋȿɃ
5.1. ɍɫɪɟɞɧɟɧɢɟ ɮɭɧɤɰɢɣ Ƚɪɢɧɚ ɩɨ ɛɟɫɩɨɪɹɞɤɭ
ɋɥɭɱɚɣɧɨ ɪɚɫɩɨɥɨɠɟɧɧɵɟ ɪɚɫɫɟɢɜɚɸɳɢɟ ɰɟɧɬɪɵ ɩɪɟɞɫɬɚɜɥɹɸɬ ɫɨɛɨɣ ɩɪɨɫɬɟɣɲɭɸ ɦɨɞɟɥɶ ɧɟɭɩɨɪɹɞɨɱɟɧɧɨɣ ɫɢɫɬɟɦɵ. ȼ ɷɬɨɣ ɝɥɚɜɟ ɦɵ ɪɚɫɫɦɨɬɪɢɦ ɬɟɯɧɢɤɭ ɭɫɪɟɞɧɟɧɢɹ ɮɭɧɤɰɢɣ Ƚɪɢɧɚ ɩɨ ɛɟɫɩɨɪɹɞɤɭ ɞɥɹ ɜɵɪɨɠɞɟɧɧɨɝɨ ɷɥɟɤɬɪɨɧɧɨɝɨ
ɝɚɡɚ ɜ ɬɪɟɯɦɟɪɧɨɣ ɫɢɫɬɟɦɟ. ɉɨɬɟɧɰɢɚɥ ɩɪɢɦɟɫɟɣ ɞɚɟɬɫɹ ɜɵɪɚɠɟɧɢɟɦ
ሺrሻ
ܷ
ൌݑ
ɝɞɟ r
ɤɨɨɪɞɢɧɚɬɵ ɰɟɧɬɪɨɜ ɩɪɢɦɟɫɟɣ, ɤɨɬɨɪɵɟ ɹɜɥɹɸɬɫɹ ɫɥɭɱɚɣɧɵɦɢ ɜɟɥɢɱɢ-
ሺ
ሻ
r െr
ǡ
ሺ
ͷǤͳ
ɧɚɦɢ ɪɚɜɧɨɦɟɪɧɨ ɪɚɫɩɪɟɞɟɥɟɧɧɵɦɢ ɩɨ ɨɛɴɟɦɭ ɨɛɪɚɡɰɚ. ɍɫɪɟɞɧɟɧɢɟ ɩɨ ɤɨɨɪɞɢɧɚɬɟ ɩɪɢɦɟɫɢ r
ɨɫɭɳɟɫɬɜɥɹɟɬɫɹ ɢɧɬɟɝɪɢɪɨɜɚɧɢɟɦ ɩɨ ɜɫɟɦɭ ɨɛɪɚɡɰɭ
ͳ
ሺǥሻ
ۦ
ۧ
ൌ
ܸ
න݀
ଷ
ሺǥሻ
ǡ
r
ሺ
ͷǤʹ
ɧɚɩɪɢɦɟɪ,
ଷ
݀
ɡɞɟɫɶ ݑ
ൌන
ሺʹߨሻ
ൻݑሺr െr
ሻ
݀
ۧ
ଷ
q
ଷ
ሻ
ൌන
ሺqሻ
Ԝݑ
ݑ൫r െr
ۦ
ݑሺr െr
ሺqሻ
ɮɭɪɶɟ-ɨɛɪɚɡ ɩɨɬɟɧɰɢɚɥɚ ݑሺr െr
q
ଷ
ሺʹߨሻ
ଷ
ሺʹߨሻ
ܸ
൯ൿൌቆ
Ԝݑ
ߜ
qr
ൻ݁
ൌ
ݑሺͲ
ܸ
ݑሺͲ
ܸ
ሻ
ቇଶǡ
ൿൌ
ሻ
ǡ
ሺqሻ
ሺqሻ
ሻ
. Ɉɱɟɜɢɞɧɨ, ɩɪɢ ɷɬɨɦ, ɱɬɨ
ሺ
ሺ
ͷǤ͵
ͷǤͶ
ɨɞɧɚɤɨ
ൌන
ଷ
݀
ሺʹߨሻ
ሺ
ۦ
ݑ
ଷ
q
݀
q
ଵ
ଶ
ሺʹߨሻ
ଷ
ൌ
Ԝݑ൫qଵ൯Ԝݑ൫qଶ൯݁
ͳ
න
ܸ
ଷ
r
ଵെr
ሺʹߨሻ
ሻݑሺ
ଷ
݀
q
q
݁
ଷ
r
ଶെr
൫q
ሺ
r
భିrమ
ሻ
r
ାq
భ
భ
ሻ
Ԝȁݑ
ۧ
ൌ
r
൯ർ݁ି൫qభାqమ൯r
మ
మ
ଶ
ሺqሻ
ȁ
Ǥ
ൌ
ሺ
ͷǤͷ
ሻ
ሻ
ሻ
ሻ
ሻ
Ƚɚɦɢɥɶɬɨɧɢɚɧ ɜɡɚɢɦɨɞɟɣɫɬɜɢɹ ɷɥɟɤɬɪɨɧɨɜ ɫ ɩɪɢɦɟɫɹɦɢ ɢɦɟɟɬ ɜɢɞ
ା
ൌන߰
ሺrሻܷሺrሻ
ܪ
ଷ
ሺrሻ
߰
݀
rǤ
43
ሺ
ͷǤ
ሻ
Ʉɚɤ ɩɨɤɚɡɚɧɨ ɜ ɤɨɧɰɟ ɝɥɚɜɵ 3, ɮɭɧɤɰɢɸ Ƚɪɢɧɚ ɞɥɹ ɷɥɟɤɬɪɨɧɨɜ ɜ ɜɧɟɲɧɟɦ
ɩɨɥɟ ܷ
ሺrሻ
, ɦɨɠɧɨ ɩɪɟɞɫɬɚɜɢɬɶ ɜ ɜɢɞɟ ɮɨɪɦɚɥɶɧɨɝɨ ɪɹɞɚ
ሺ
ሻ
͵Ǥ͵Ͷ
, ɤɨɬɨɪɵɣ ɞɥɹ
ɭɞɨɛɫɬɜɚ ɦɵ ɩɟɪɟɩɢɲɟɦ ɟɳɟ ɪɚɡ
ሺ
ሻ
ܷ
r
ܩሺ߱ǡrǡr
න݀
ᇱ
ሻ
ሺ
ൌܩ
߱ǡr െr
ଷ
ଷ
ሺ
r
ଶܩ
߱ǡr െr
r
݀
ଵ
ᇱ
ሻ
ሺ
ܷ
ሻ
ଵ
න݀
ሻ
r
ଵ
ܩ
ଷ
ሺ
߱ǡr െr
r
ଵܩ
ሺ
ሺ
߱ǡr
ଵെrଶ
ܷ
ሻ
ଵ
ሻ
ଵ
ሻ
r
ଶ
ܩ
ሺ
߱ǡr
ܩ
ሺ
߱ǡr
െr
ଶ
ᇱ
ሻ
െr
ଵ
ᇱ
ሻ
ڮǤሺͷǤ
Ʉɚɠɞɨɦɭ ɫɥɚɝɚɟɦɨɦɭ ɷɬɨɝɨ ɪɹɞɚ ɦɨɠɧɨ ɩɨɫɬɚɜɢɬɶ ɜ ɫɨɨɬɜɟɬɫɬɜɢɟ ɞɢɚɝɪɚɦɦɭ
(ɪɢɫ. 5.1). ȼ ɤɚɠɞɨɣ ɬɚɤɨɣ ɞɢɚɝɪɚɦɦɟ ɫɩɥɨɲɧɨɣ ɥɢɧɢɢ ɫɨ ɫɬɪɟɥɤɨɣ ɫɨɨɬɜɟɬɫɬɜɭɟɬ
ɮɭɧɤɰɢɹ Ƚɪɢɧɚ ܩ
ሺ߱ǡܚଵെܚଶሻ, ɚ ɤɚɠɞɨɦɭ ɤɪɟɫɬɢɤɭ ɫɨɨɬɜɟɬɫɬɜɭɟɬ ɩɨɬɟɧɰɢɚɥ ɩɪɢ-
ɦɟɫɟɣ ܷሺ࢟ሻȀ. ȼ ɨɤɨɧɱɚɬɟɥɶɧɨɦ ɜɵɪɚɠɟɧɢɢ ɞɥɹ ɞɢɚɝɪɚɦɦɵ ɩɪɨɜɨɞɢɬɫɹ ɢɧɬɟɝɪɢ-
ɪɨɜɚɧɢɟ ɩɨ ɜɫɟɦ ɤɨɨɪɞɢɧɚɬɚɦ ɩɨɬɟɧɰɢɚɥɨɜ ɩɪɢɦɟɫɟɣ.
Ɋɢɫ. 5.1
ሻ
Ⱦɥɹ ɭɫɪɟɞɧɟɧɢɹ ɮɭɧɤɰɢɢ Ƚɪɢɧɚ ɩɨ ɛɟɫɩɨɪɹɞɤɭ, ɡɚɦɟɬɢɦ, ɱɬɨ ɩɨɬɟɧɰɢɚɥ ܷ
ሺrሻ
ɜ (5.1) ɟɫɬɶ ɫɭɦɦɚ ɧɟɡɚɜɢɫɢɦɵɯ ɜɤɥɚɞɨɜ ɨɬɞɟɥɶɧɵɯ ɩɪɢɦɟɫɟɣ. ɋɨɨɬɜɟɬɫɬɜɟɧɧɨ,
ɜɟɪɲɢɧɚɦ ɧɚ ɪɢɫ. 5.1 ɞɨɥɠɧɵ ɛɵɬɶ ɩɪɢɩɢɫɚɧɵ ɧɨɦɟɪɚ
ɩɪɢɦɟɫɟɣ, ɩɨ ɤɨɬɨɪɵɦ ɡɚ-
ɬɟɦ ɧɚɞɨ ɩɪɨɫɭɦɦɢɪɨɜɚɬɶ (ɪɢɫ. 5.2).
Ɋɢɫ. 5.2
ȼɧɚɱɚɥɟ ɪɚɫɫɦɨɬɪɢɦ ɞɢɚɝɪɚɦɦɵ, ɜ ɤɨɬɨɪɵɯ ɜɫɟ ɧɨɦɟɪɚ ɩɪɢɦɟɫɟɣ ɪɚɡɧɵɟ. ɉɨɫɥɟ ɭɫɪɟɞɧɟɧɢɹ ɩɨ ɩɨɥɨɠɟɧɢɸ ɩɪɢɦɟɫɟɣ ɫ ɭɱɟɬɨɦ ɢɯ ɫɬɚɬɢɫɬɢɱɟɫɤɨɣ ɧɟɡɚɜɢɫɢɦɨɫɬɢ ɢ ɪɚɜɟɧɫɬɜɚ (5.3) ɩɨɥɭɱɢɦ, ɱɬɨ ɤɚɠɞɚɹ ɜɟɪɲɢɧɚ ɩɨɫɥɟ ɫɭɦɦɢɪɨɜɚɧɢɹ ɞɚɫɬ
ɦɧɨɠɢɬɟɥɶ ݊
ɫɞɜɢɝɭ ɯɢɦɢɱɟɫɤɨɝɨ ɩɨɬɟɧɰɢɚɥɚ ɜ ɮɭɧɤɰɢɢ Ƚɪɢɧɚ ɧɚ ɜɟɥɢɱɢɧɭ ߜߤൌ݊
ݑሺͲሻȀ, ɝɞɟ ݊ൌܰȀܸ ɤɨɧɰɟɧɬɪɚɰɢɹ ɩɪɢɦɟɫɟɣ. ɗɬɨ ɷɤɜɢɜɚɥɟɧɬɧɨ
ݑሺͲ
ሻ
, ɱɬɨ
ɟɫɬɟɫɬɜɟɧɧɨ, ɩɨɬɨɦɭ ɱɬɨ ɷɬɚ ɜɟɥɢɱɢɧɚ ɪɚɜɧɚ ɫɪɟɞɧɟɦɭ ɩɨɬɟɧɰɢɚɥɭ ɩɪɢɦɟɫɟɣ, ɪɚɡɦɚɡɚɧɧɵɯ ɩɨ ɨɛɴɟɦɭ. Ɇɨɠɧɨ ɩɨɷɬɨɦɭ ɫɪɚɡɭ ɞɨɛɚɜɢɬɶ ɤ ɯɢɦɢɱɟɫɤɨɦɭ ɩɨɬɟɧɰɢɚɥɭ
ɜɟɥɢɱɢɧɭ െߜߤ ɢ ɫɱɢɬɚɬɶ, ɱɬɨ ɜɫɟ ɬɚɤɢɟ ɞɢɚɝɪɚɦɦɵ ɫ ɭɫɪɟɞɧɟɧɢɟɦ ɩɨ ɩɨɥɨɠɟɧɢɸ
ɨɞɢɧɨɱɧɵɯ ɩɪɢɦɟɫɟɣ ɞɚɸɬ ɧɭɥɟɜɨɣ ɜɤɥɚɞ. ɉɨ-ɞɪɭɝɨɦɭ ɷɬɨɝɨ ɠɟ ɪɟɡɭɥɶɬɚɬɚ ɦɨɠɧɨ
ɞɨɛɢɬɶɫɹ, ɫɱɢɬɚɹ ɫɪɟɞɧɟɟ ɡɧɚɱɟɧɢɟ
ɩɪɢɦɟɫɧɨɝɨ ɩɨɬɟɧɰɢɚɥɚ ɪɚɜɧɨɦɭ ɧɭɥɸ:
ሻ
ۦ
ۧ
ൌͲǤ
ܷሺ࢘
ሺ
ͷǤͺ
Ⱦɚɥɟɟ ɪɚɫɫɦɨɬɪɢɦ ɞɢɚɝɪɚɦɦɵ, ɜ ɤɨɬɨɪɵɯ ɟɫɬɶ ɩɚɪɚ ɜɟɪɲɢɧ ɫ ɨɞɧɢɦ ɢ ɬɟɦ ɠɟ
ɧɨɦɟɪɨɦ ɩɪɢɦɟɫɢ. ɑɬɨɛɵ ɨɬɦɟɬɢɬɶ ɷɬɨɬ ɮɚɤɬ, ɫɨɟɞɢɧɢɦ ɲɬɪɢɯɨɜɨɣ ɥɢɧɢɟɣ ɜɟɪɲɢɧɵ, ɨɬɜɟɱɚɸɳɢɟ ɫɨɜɩɚɞɚɸɳɢɦ ɩɪɢɦɟɫɹɦ (ɪɢɫ. 5.3). ɍɫɪɟɞɧɟɧɢɟ ɩɨ ɩɨɥɨɠɟɧɢɸ ɷɬɨɣ ɩɪɢɦɟɫɢ, ɜɯɨɞɹɳɟɣ ɜ ɞɜɟ ɪɚɡɧɵɟ ɜɟɪɲɢɧɵ, ɢ ɡɚɬɟɦ ɫɭɦɦɢɪɨɜɚɧɢɟ ɩɨ
ɱɢɫɥɭ ɬɚɤɢɯ ɩɪɢɦɟɫɟɣ, ɚ
ɢɯ ɱɢɫɥɨ ܰ
, ɞɚɟɬ ɫ ɭɱɟɬɨɦ (5.5)
44
ሻ
ଷ
݀
න
ሺʹߨሻ
q
݁
ଷ
ሻ
ሺ
q
r
భିrమ
Ԝȁݑ
ሺqሻ
ଶ
ȁ
ǡ
ሺ
ͷǤͻ
ሻ
݊
ଶ
ɡɞɟɫɶ r
ɢ rଶ ɤɨɨɪɞɢɧɚɬɵ ɜɟɪɲɢɧ.
ଵ
Ɋɢɫ. 5.3
Ɉɛɨɡɧɚɱɢɦ ɷɬɨ ɜɵɪɚɠɟɧɢɟ ߨ
ሺ
r
ଵെrଶ
ሻ
ɢ ɛɭɞɟɦ ɪɚɫɫɦɚɬɪɢɜɚɬɶ ɲɬɪɢɯɨɜɭɸ ɥɢɧɢɸ ɤɚɤ ɧɨɜɵɣ ɷɥɟɦɟɧɬ ɞɢɚɝɪɚɦɦɧɨɣ ɬɟɯɧɢɤɢ, ɩɨ ɬɪɚɞɢɰɢɢ ɟɟ ɧɚɡɵɜɚɸɬ ɤɪɟɫɬɨɜɨɣ ɞɢɚɝɪɚɦɦɧɨɣ ɬɟɯɧɢɤɨɣ. Ɏɨɪɦɚɥɶɧɨ ɨɧɚ ɧɚɩɨɦɢɧɚɟɬ ɞɢɚɝɪɚɦɦɧɭɸ ɬɟɯɧɢɤɭ
ɞɥɹ ɷɥɟɤɬɪɨɧ-ɮɨɧɨɧɧɨɝɨ ɜɡɚɢɦɨɞɟɣɫɬɜɢɹ, ɫ ɡɚɦɟɧɨɣ ɮɭɧɤɰɢɢ Ƚɪɢɧɚ ɮɨɧɨɧɨɜ ɧɚ
ɩɪɢɦɟɫɧɭɸ ɥɢɧɢɸ ɫ ɚɧɚɥɨɝɨɦ ɮɭɧɤɰɢɢ Ƚɪɢɧɚ, ɪɚɜɧɵɦ ߨ
ሺ
r
ଵെrଶ
ሻ
. ɇɨ ɩɪɢ ɷɬɨɦ
ɢɦɟɸɬɫɹ ɞɜɚ ɫɭɳɟɫɬɜɟɧɧɵɯ ɨɬɥɢɱɢɹ.
ȼɨ-ɩɟɪɜɵɯ, ɩɟɪɟɞɚɱɚ ɷɧɟɪɝɢɢ ɩɨ ɲɬɪɢɯɨɜɨɣ ɥɢɧɢɢ ɪɚɜɧɚ ɧɭɥɸ (ɭɩɪɭɝɨɟ ɪɚɫ-
ɫɟɹɧɢɟ).
ȼɨ-ɜɬɨɪɵɯ, ɨɬɫɭɬɫɬɜɭɸɬ ɩɨɥɹɪɢɡɚɰɢɨɧɧɵɟ ɩɟɬɥɢ, ɨɞɟɜɚɸɳɢɟ ɲɬɪɢɯɨɜɵɟ
ɥɢɧɢɢ.
ɉɟɪɜɨɟ ɨɬɥɢɱɢɟ ɨɱɟɜɢɞɧɨ, ɚ ɜ ɫɩɪɚɜɟɞɥɢɜɨɫɬɢ ɜɬɨɪɨɝɨ ɥɟɝɤɨ ɭɛɟɞɢɬɶɫɹ, ɩɪɨɫɥɟɞɢɜ ɤɚɤ ɜ ɞɢɚɝɪɚɦɦɚɯ ɪɢɫ. 5.1 ɩɪɢ ɭɫɪɟɞɧɟɧɢɢ ɩɨɹɜɥɹɸɬɫɹ ɩɪɢɦɟɫɧɵɟ ɥɢɧɢɢ.
Ɏɚɤɬɢɱɟɫɤɢ
ɡɚɦɤɧɭɬɵɟ ɩɟɬɥɢ ɨɬɫɭɬɫɬɜɭɸɬ ɢɡ-ɡɚ ɬɨɝɨ, ɱɬɨ ɩɨɬɟɧɰɢɚɥ ɩɪɢɦɟɫɟɣ
«ɜɦɨɪɨɠɟɧɧɵɣ» ɢ ɩɨɥɨɠɟɧɢɟ ɩɪɢɦɟɫɟɣ ɧɟ ɩɨɞɫɬɪɚɢɜɚɟɬɫɹ ɩɨɞ ɡɚɞɚɧɧɭɸ ɷɥɟɤ-
ɬɪɨɧɧɭɸ ɩɥɨɬɧɨɫɬɶ.
Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, ɞɥɹ ɭɫɪɟɞɧɟɧɢɹ ɮɭɧɤɰɢɢ Ƚɪɢɧɚ ɩɨ ɩɨɥɨɠɟɧɢɸ ɩɪɢɦɟɫɟɣ
ɧɭɠɧɨ ɜ ɤɚɠɞɨɣ ɞɢɚɝɪɚɦɦɟ ɫ ɱɟɬɧɵɦ ɱɢɫɥɨɦ ɜɟɪɲɢɧ ɪɢɫ. 5.3 ɪɚɡɛɢɬɶ ɷɬɢ ɜɟɪɲɢɧɵ ɧɚ ɩɚɪɵ ɢ ɜ ɤɚɠɞɨɣ ɩɚɪɟ ɫɨɟɞɢɧɢɬɶ
ɲɬɪɢɯɨɜɨɣ ɥɢɧɢɟɣ ɟɟ ɜɟɪɲɢɧɵ. ɗɬɨ
ɛɭɞɟɬ ɜɤɥɚɞ ɜ ɭɫɪɟɞɧɟɧɧɭɸ ɮɭɧɤɰɢɸ Ƚɪɢɧɚ.
ɇɚɩɪɢɦɟɪ, ɞɥɹ ɩɟɪɜɨɣ ɧɟ ɢɫɱɟɡɚɸɳɟɣ ɞɢɚɝɪɚɦɦɵ ɩɪɨɰɟɫɫ ɭɫɪɟɞɧɟɧɢɹ ɞɚɟɬ
ଷ
න݀
r
՜න݀
ሺ
ሻ
ܷ
r
ଷ
ሺ
r
r
ଶܩ
݀
ଵ
ଷ
r
߱ǡr െr
ሺ
ଶܩ
߱ǡr െr
݀
ଵ
ଷ
ଵ
ሻ
ଵ
ଵ
ሻߨሺ
ܩ
r
ଵെrଶ
ሺ
߱ǡr
ଵെrଶ
ሻ
ܩ
ሺ
߱ǡr
ܷ
ሻ
ଵെrଶ
ሺ
ሻ
r
ଶ
ሺ
߱ǡr
ܩ
ሻ
ሺ
ܩ
߱ǡr
ᇱ
ሻ
՜
െr
ଶ
ᇱ
ሻǡሺ
െr
ଶ
ͷǤͳͲ
ɩɨɥɭɱɟɧɧɨɟ ɜɵɪɚɠɟɧɢɟ, ɤɚɤ ɢ ɫɥɟɞɨɜɚɥɨ ɨɠɢɞɚɬɶ, ɬɪɚɧɫɥɹɰɢɨɧɧɨ-ɢɧɜɚɪɢɚɧɬɧɨ ɢ
ɜ ɢɦɩɭɥɶɫɧɨɦ ɩɪɟɞɫɬɚɜɥɟɧɢɢ ɢɦɟɟɬ ɜɢɞ
ଷ
݀
ܩ
ሺ
߱ǡpሻቈ
݊
p
ଶ
න
ଵ
หݑ൫p െpଵ൯หଶܩ൫߱ǡpଵ൯ܩ
ଷ
ሺʹߨሻ
ሺ
߱ǡpሻǤ
ሺ
ͷǤͳͳ
ɗɬɨɦɭ ɜɵɪɚɠɟɧɢɸ ɫɨɨɬɜɟɬɫɬɜɭɟɬ ɞɢɚɝɪɚɦɦɚ ɪɢɫ. 5.4.
45
ሻ
ሻ
Ɋɢɫ. 5.4
ȼ ɫɥɟɞɭɸɳɟɦ ɩɨɪɹɞɤɟ ɛɭɞɟɬ ɱɟɬɵɪɟ ɜɟɪɲɢɧɵ, ɢ ɢɯ ɪɚɡɛɢɬɶ ɧɚ ɩɚɪɵ ɦɨɠɧɨ
ɬɪɟɦɹ ɫɩɨɫɨɛɚɦɢ, ɜ ɢɬɨɝɟ ɜ ɢɦɩɭɥɶɫɧɨɦ ɩɪɟɞɫɬɚɜɥɟɧɢɢ ɛɭɞɟɬ ɬɪɢ ɫɭɳɟɫɬɜɟɧɧɨ
ɪɚɡɧɵɯ ɜɤɥɚɞɚ (ɪɢɫ. 5.5 (a) ɢ 5.5 (ɛ)).
Ɋɢɫ. 5.5 (ɚ) Ɋɢɫ. 5.5 (ɛ)
ɉɟɪɜɵɣ ɜɤɥɚɞ ɫɨɨɬɜɟɬɫɬɜɭɟɬ ɞɢɚɝɪɚɦɦɟ, ɜ ɤɨɬɨɪɨɣ ɩɪɢɦɟɫɧɵɟ ɥɢɧɢɢ ɧɟ ɩɟɪɟɫɟɤɚɸɬɫɹ ɢ
ɧɟ ɨɯɜɚɬɵɜɚɸɬ ɞɪɭɝ ɞɪɭɝɚ
ଷ
ଶ
݀
ሾ
ܩ
ሺ
߱ǡp
ൈቈන
ଶ
ሻ
ሿ
ଷ
݀
ሺʹߨሻ
݊
ସ
p
ଶ
ଷ
p
ሺʹߨሻ
ଵ
ଷ
ቈන
หݑ൫p െpଶ൯หଶܩ൫߱ǡpଶ൯ܩ
หݑ൫p െpଵ൯หଶܩ൫߱ǡpଵ൯ൈ
ሺ
߱ǡpሻǡ
ሺ
ͷǤͳʹ
ɜɨ ɜɬɨɪɨɣ ɞɢɚɝɪɚɦɦɟ ɭɫɥɨɠɧɹɟɬɫɹ ɷɥɟɤɬɪɨɧɧɚɹ ɥɢɧɢɹ, ɧɚɯɨɞɹɳɚɹɫɹ ɩɨɞ ɩɟɪɜɨɣ
ɩɪɢɦɟɫɧɨɣ ɥɢɧɢɟɣ ɢ ɜɵɪɚɠɟɧɢɟ ɞɥɹ ɷɬɨɣ ɞɢɚɝɪɚɦɦɵ ɦɵ ɨɩɭɫɬɢɦ. Ⱦɢɚɝɪɚɦɦɚ ɫ
ɩɟɪɟɫɟɱɟɧɢɹɦɢ (ɪɢɫ. 5.5 (ɜ)) ɞɚɟɬ
ଷ
ଶ
݀
݊
ଶ
ሺ
ሻ
߱ǡp
ሿ
න
ସ
ሺʹߨሻ
ሾ
ܩ
ଷ
p
݀
p
ଵ
ଶ
ଷ
หݑ൫p െpଵ൯หଶหݑ൫pଶെp൯หଶܩ൫߱ǡpଵ൯ܩ൫߱ǡpଶ൯ൈ
ଷ
ሺʹߨሻ
ሻ
൫߱ǡpଵpଶെp൯Ǥ
ൈܩ
Ɋɢɫ. 5.5 (ɜ)
46
ሺ
ͷǤͳ͵
ሻ
ɇɢɠɟ ɦɵ ɩɨɤɚɠɟɦ, ɱɬɨ ɜ ɪɚɫɫɦɚɬɪɢɜɚɟɦɨɦ ɫɥɭɱɚɟ (ɬɪɟɯɦɟɪɧɵɣ ɜɵɪɨɠɞɟɧɧɵɣ ɮɟɪɦɢ-ɝɚɡ) ɜɫɟ ɞɢɚɝɪɚɦɦɵ ɫ ɩɟɪɟɫɟɱɟɧɢɟɦ ɨɤɚɡɵɜɚɸɬɫɹ ɦɚɥɵ ɩɨ ɫɪɚɜɧɟɧɢɸ ɫ ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɦɢ ɞɢɚɝɪɚɦɦɚɦɢ ɛɟɡ ɩɟɪɟɫɟɱɟɧɢɣ, ɝɞɟ ɢ ɭɤɚɠɟɦ ɫɬɟɩɟɧɶ
ɦɚɥɨɫɬɢ.
Ⱦɢɚɝɪɚɦɦɵ ɛɟɡ ɩɟɪɟɫɟɱɟɧɢɣ ɦɨɠɧɨ ɩɪɨɫɭɦɦɢɪɨɜɚɬɶ. ȼ ɪɟɡɭɥɶɬɚɬɟ ɷɬɨɝɨ ɫɭɦ-
ɦɢɪɨɜɚɧɢɹ ɩɨɥɭɱɢɦ ɬɨɱɧɭɸ ɝɪɢɧɨɜɫɤɭɸ ɮɭɧɤɰɢɸ ܩሺ߱ǡp
ሻ
, ɤɨɬɨɪɨɣ ɜ
ɞɢɚɝɪɚɦ-
ɦɚɯ ɫɨɨɬɜɟɬɫɬɜɭɟɬ ɬɨɥɫɬɚɹ ɥɢɧɢɹ. ɍɪɚɜɧɟɧɢɟ ɧɚ ɷɬɭ ɮɭɧɤɰɢɸ ɦɨɠɧɨ ɩɨɥɭɱɢɬɶ
ɞɢɚɝɪɚɦɦɧɵɦ ɫɩɨɫɨɛɨɦ.
Ⱦɥɹ ɷɬɨɝɨ ɧɭɠɧɨ ɜ ɞɢɚɝɪɚɦɦɟ ɫ ɨɞɧɨɣ ɩɪɢɦɟɫɧɨɣ ɥɢɧɢɟɣ (ɪɢɫ. 5.4), ɭɬɨɥɳɢɬɶ,
(ɬɨ ɟɫɬɶ ɫɞɟɥɚɬɶ ɡɚɦɟɧɭ ܩ
ɧɨɟ ɜɵɪɚɠɟɧɢɟ ɜɦɟɫɬɟ ɫɨ ɫɥɚɝɚɟɦɵɦ ܩ
՜ܩ) ɜɬɨɪɭɸ ɢ ɬɪɟɬɶɸ ɷɥɟɤɬɪɨɧɧɵɟ ɥɢɧɢɢ. ɉɨɥɭɱɟɧ-
ሺ
߱ǡpሻ ɜ ɬɨɱɧɨɫɬɢ ɪɚɜɧɨ ɮɭɧɤɰɢɢ ܩሺ߱ǡp
(ɪɢɫ. 5.6).
Ɋɢɫ. 5.6
ȼ ɚɧɚɥɢɬɢɱɟɫɤɨɦ ɜɢɞɟ ɷɬɨ ɭɪɚɜɧɟɧɢɟ ɢɦɟɟɬ ɜɢɞ
ሻ
ܩሺ߱ǡpሻൌܩ
ሺ
߱ǡpሻܩ
ሺ
߱ǡpሻԜߑሺ߱ǡpሻԜܩሺ߱ǡpሻǡ
ሺ
ͷǤͳͶ
ɝɞɟ
ଷ
݀
ߑሺ߱ǡpሻൌ
݊
p
ଶ
න
ଵ
หݑ൫p െpଵ൯หଶܩ൫߱ǡpଵ൯Ǥ
ଷ
ሺʹߨሻ
ሺ
ͷǤͳͷ
ɂɡ ɭɪɚɜɧɟɧɢɹ (5.14) ɩɨɥɭɱɢɦ
ିଵ
ܩ
ሺ
߱ǡpሻൌܩ
ିଵ
ሺ
߱ǡpሻെߑሺ߱ǡpሻൌ߱െߦ
െߑሺ߱ǡp
p
ሻሺ
ͷǤͳ
ɢɥɢ
ܩሺ߱ǡpሻൌ
߱െߦ
ͳ
െߑሺ߱ǡp
p
ǡ
ሻ
ɩɨɞɫɬɚɜɥɹɟɦ ɷɬɨ ɜ (5.15), ɩɨɥɭɱɢɦ ɢɧɬɟɝɪɚɥɶɧɨɟ ɭɪɚɜɧɟɧɢɟ ɧɚ ߑሺ߱ǡp
ଶ
Ǥ
ሻ
ߑሺ߱ǡpሻൌ
ଷ
݀
p
݊
න
ଶ
ሺʹߨሻ
หݑ൫p െpଵ൯ห
ଵ
ଷ
߱െߦpെߑሺ߱ǡp
ሺ
ͷǤͳ
ሻ
:
ሺ
ͷǤͳͺ
ɗɬɨ ɭɪɚɜɧɟɧɢɟ ɧɭɠɧɨ ɪɟɲɚɬɶ ɩɨ ɨɬɞɟɥɶɧɨɫɬɢ ɞɥɹ ɡɚɩɚɡɞɵɜɚɸɳɟɣ ɢ ɨɩɟɪɟɠɚɸɳɟɣ ɮɭɧɤɰɢɢ Ƚɪɢɧɚ. Ⱦɥɹ ɡɚɩɚɡɞɵɜɚɸɳɟɣ ɮɭɧɤɰɢɢ ɜ ɩɟɪɜɨɦ ɩɪɢɛɥɢɠɟɧɢɢ ɜ
ɩɪɚɜɨɣ ɱɚɫɬɢ ɧɭɠɧɨ ɩɨɥɨɠɢɬɶ
47
ሻ
ሻ
ሻ
ሻ
ሻ
ɬɨɝɞɚ
ோ
ߑ
ଵ
ோ
ሺ
߱ǡpሻൌെ݅ߜǡԜԜԜߜ՜Ͳǡ
ߑ
݊
ሺ
߱ǡpሻൌ
ଶ
න
ଷ
݀
ሺʹߨሻ
p
หݑ൫p െpଵ൯ห
ଵ
ଷ
߱െߦp݅ߜ
ሺ
ሻ
ͷǤͳͻ
ଶ
ሺ
ǡ
ͷǤʹͲ
ሻ
ɩɟɪeɣɞɟɦ ɜ ɢɧɬɟɝɪɢɪɨɜɚɧɢɢ ɩɨ p
ଷ
݀
ଵൌʹߨߠ݀ߠԜ
ൌଵԜn
ଵ
ɤ ɫɮɟɪɢɱɟɫɤɢɦ ɤɨɨɪɞɢɧɚɬɚɦ
ଵ
ଶ
݀ଵൌ݀ߗ
ଵ
ଶ
݀ଵǡ
n
ଵ
భ
ɞɚɥɟɟ ɩɨɥɨɠɢɦ
ɝɞɟ Ԝ
ɬɪɨɧɨɜ ݊
ɢɦɩɭɥɶɫ Ɏɟɪɦɢ ɫɢɫɬɟɦɵ ɷɥɟɤɬɪɨɧɨɜ, ɫɜɹɡɚɧɧɵɣ ɫ ɤɨɧɰɟɧɬɪɚɰɢɟɣ ɷɥɟɤ-
ɫɨɨɬɧɨɲɟɧɢɟɦ (ɞɥɹ ɩɪɨɫɬɨɬɵ ɫɱɢɬɚɟɦ ɷɥɟɤɬɪɨɧɵ ɛɟɫɫɩɢɧɨɜɵɦɢ)
Ͷߨ
͵
ଵൌ
ቀ
ቁଷൌ݊Ǥ
ʹߨ
݇ǡ
Ʉɚɤ ɦɵ ɭɜɢɞɢɦ, ɝɥɚɜɧɵɣ ɜɤɥɚɞ ɜ ɢɧɬɟɝɪɚɥɵ ɛɭɞɟɬ ɜɧɨɫɢɬ ɨɛɥɚɫɬɶ
ɩɨɷɬɨɦɭ
ଵൌ
݇ሻn
ଶ
Ԝ݀ߗ
݀݇Ǥ
n
భ
݀
ɍɱɬɟɦ ɬɚɤɠɟ, ɱɬɨ ɩɪɢ p ൌሺ
ߦ
ൌݒ݇ǡԜԜԜݒൌȀ݉ǡ
p
ଷ
ɢ ɩɨɷɬɨɦɭ ɦɨɠɧɨ ɩɟɪɟɣɬɢ ɤ ɢɧɬɟɝɪɢɪɨɜɚɧɢɸ ɩɨ ߦ ɜɦɟɫɬɨ ݇
ଶ
݉
ݒ
݀
ଷ
ଵൌ
Ԝ݀ߗ
݀ߦǤ
n
ଶ
భ
ȼ ɢɬɨɝɟ
ଶ
൯ห
ଵ
ǡ
ோ
ߑ
ଵ
ሺ
߱ǡpሻൌ
݊
݉ଶݒ
ሺʹߨሻ
ଷସ
න݀ߗ
݀ߦ
n
భ
หݑ൫p െp
߱െߦ݅ߜ
ȁ݇ȁ
ሺ
ͷǤʹͳ
ሺ
ͷǤʹʹ
ሺ
ͷǤʹ͵
ا
ሺ
ͷǤʹͶ
ሺ
ͷǤʹͷ
ሺ
ͷǤʹ
ሺ
ͷǤʹ
ሻ
ሻ
ሻ
,
ሻ
ሻ
ሻ
ሻ
ɜɟɳɟɫɬɜɟɧɧɚɹ ɱɚɫɬɶ ɷɬɨɝɨ ɢɧɬɟɝɪɚɥɚ ɧɟ ɩɪɟɞɫɬɚɜɥɹɟɬ ɢɧɬɟɪɟɫɚ, ɬɚɤ ɤɚɤ ɹɜɥɹɟɬɫɹ
ɩɟɪɟɧɨɪɦɢɪɨɜɤɨɣ ɯɢɦɢɱɟɫɤɨɝɨ ɩɨɬɟɧɰɢɚɥɚ ɜɨ ɜɬɨɪɨɦ ɩɨɪɹɞɤɟ, ɚ ɞɥɹ ɦɧɢɦɨɣ ɱɚɫɬɢ ɢɦɟɟɦ
݊
݉ଶݒ
ோ
ሺ
߱ǡpሻൌെ݅ߨ
ߑ
ଵ
ൌെ݅
ሺʹߨሻ
݊
݉ଶݒ
Ͷߨ
ଷସ
ସ
න݀ߗ
గ
න݀ߠ
݀ߦหݑ൫p െpଵ൯หଶߜሺ߱െߦሻൌ
n
భ
ߠหݑ൫p െpଵ൯หଶǡ
ሺ
ͷǤʹͺ
48
ሻ
ȁ
ɝɞɟ ȁp
หൌ, ɚ ߠ ɟɫɬɶ ɭɝɨɥ ɦɟɠɞɭ ɜɟɤɬɨɪɚɦɢ p ɢ p
ൌหp
ଵ
. ɍɱɬɟɦ ɬɟɩɟɪɶ ɮɨɪɦɭɥɭ
ଵ
ɞɥɹ ɩɨɥɧɨɝɨ ɫɟɱɟɧɢɹ ɪɚɫɫɟɹɧɢɹ ߪ ɜ ɛɨɪɧɨɜɫɤɨɦ ɩɪɢɛɥɢɠɟɧɢɢ ɧɚ ɩɨɬɟɧɰɢɚɥɟ
ሺrሻ
ݑ
(ɫɦ. [1], ɫɬɪ. 598):
గ
ଶ
ߪൌ
݉
ʹߨ
න݀ߠ
ସ
ߠหݑ൫p െpଵ൯หଶǡ
ሺ
ͷǤʹͻ
ɬɨɝɞɚ ɦɨɠɧɨ ɡɚɩɢɫɚɬɶ
ோ
ሺ
߱ǡpሻൌെ݅Ȁʹ߬ǡ
ߑ
ଵ
ሺ
ͷǤ͵Ͳ
ɝɞɟ ɜɟɥɢɱɢɧɚ ߬ ɬɚɤɚɹ, ɱɬɨ
ͳȀ߬ൌ݊
ݒ
ߪǡ
ሺ
ͷǤ͵ͳ
ɢɦɟɟɬ ɪɚɡɦɟɪɧɨɫɬɶ ɜɪɟɦɟɧɢ ɢ ɦɨɠɟɬ ɛɵɬɶ ɢɧɬɟɪɩɪɟɬɢɪɨɜɚɧɚ ɤɚɤ ɫɪɟɞɧɟɟ ɜɪɟɦɹ
ɧɚɯɨɠɞɟɧɢɹ ɷɥɟɤɬɪɨɧɚ ɜ ɫɨɫɬɨɹɧɢɢ ɫ ɢɦɩɭɥɶɫɨɦ p.
ோ
ሺ
ሻ
߱ǡp
Ɍɟɩɟɪɶ ɞɥɹ ɮɭɧɤɰɢɢ ߑ
ோ
ሺ
߱ǡpሻൌ
ߑ
ଶ
ଶ
ɜ ɫɥɟɞɭɸɳɟɦ ɩɪɢɛɥɢɠɟɧɢɢ ɢɦɟɟɦ
݊
݉ଶݒ
ሺʹߨሻ
ଷସ
න݀ߗ
หݑ൫p െp
݀ߦ
n
భ
߱െߦ݅Ȁʹ߬
൯ห
ଵ
ଶ
ǡ
ሺ
ͷǤ͵ʹ
ɩɪɟɞɩɨɥɚɝɚɹ, ɱɬɨ ɜɪɟɦɹ ߬ ɭɞɨɜɥɟɬɜɨɪɹɟɬ ɭɫɥɨɜɢɸ
ݒ
߬بͳǡ
(ɨɡɧɚɱɚɸɳɟɦɭ, ɱɬɨ ɞɥɢɧɚ ɫɜɨɛɨɞɧɨɝɨ ɩɪɨɛɟɝɚ ɷɥɟɤɬɪɨɧɚ ݒ
ɯɚɪɚɤɬɟɪɧɨɣ ɞɥɢɧɵ ɜɨɥɧɵ Ⱦɟ Ȼɪɨɣɥɹ
ɪɟɡɭɥɶɬɚɬ, ɱɬɨ ɢ ɜ ɩɟɪɜɨɦ ɩɪɢɛɥɢɠɟɧɢɢ. Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, ɢɬɟɪɚɰɢɨɧɧɵɣ ɩɪɨɰɟɫɫ
ோ
ɫɪɚɡɭ ɫɯɨɞɢɬɫɹ, ɢ ߑ
ɨɬɜɟɬ ɞɥɹ ߑ
ோ
ሺ
߱ǡpሻൌߑ
ோ
ሺ
ଵ
ோ
ߑ
ିଵ
), ɜɵɱɢɫɥɟɧɢɟ ɢɧɬɟɝɪɚɥɚ ɞɚɟɬ ɬɨɬ ɠɟ
ሻ
߱ǡp
. ȼ ɪɟɡɭɥɶɬɚɬɟ ɩɨɥɭɱɚɟɦ ɨɤɨɧɱɚɬɟɥɶɧɵɣ
ሺ
߱ǡpሻൌെ݅Ȁʹ߬ǡ
߬ ɧɚɦɧɨɝɨ ɛɨɥɶɲɟ ɟɝɨ
ሺ
ͷǤ͵͵
ሺ
ͷǤ͵Ͷ
ሻ
ሻ
ሻ
ሻ
ሻ
ሻ
ɝɞɟ
ͳ
߬
݉
ݒ
ൌ
ʹߨ
ସ
න݀ߠ
ߠหݑ൫p െpଵ൯หଶǤ
గ
ଶ
݊
ɗɬɨ ɧɚɦ ɞɚɟɬ ɭɫɪɟɞɧɟɧɧɭɸ ɩɨ ɩɨɥɨɠɟɧɢɸ ɩɪɢɦɟɫɟɣ ɡɚɩɚɡɞɵɜɚɸɳɭɸ ɮɭɧɤɰɢɸ Ƚɪɢɧɚ
߱െߦ
ͳ
݅Ȁʹ߬
p
Ǥ
ோ
ܩ
ሺ
߱ǡpሻൌ
Ⱦɥɹ ɨɩɟɪɟɠɚɸɳɟɣ ɮɭɧɤɰɢɢ ɬɨɱɧɨ ɬɚɤɢɟ ɠɟ ɜɵɱɢɫɥɟɧɢɹ ɞɚɞɭɬ
߱െߦ
ͳ
െ݅Ȁʹ߬
p
Ǥ
ܩ
ሺ
߱ǡpሻൌ
49
ሺ
ͷǤ͵ͷ
ሺ
ͷǤ͵
ሺ
ͷǤ͵
ሻ
ሻ
ሻ
ɉɨɤɚɠɟɦ ɬɟɩɟɪɶ ɧɚ ɩɪɢɦɟɪɟ ɞɢɚɝɪɚɦɦ, ɩɪɟɞɫɬɚɜɥɟɧɧɵɯ ɧɚ ɪɢɫ. 5.5, ɱɬɨ ɞɢɚɝɪɚɦɦɚ ɫ ɩɟɪɟɫɟɱɟɧɢɟɦ ɩɪɢɦɟɫɧɵɯ ɥɢɧɢɣ ɦɚɥɚ ɩɨ ɨɬɧɨɲɟɧɢɸ ɤ ɞɢɚɝɪɚɦɦɟ ɛɟɡ
ɩɟɪɟɫɟɱɟɧɢɣ. Ʉɚɤ ɫɥɟɞɭɟɬ ɢɡ (5.36), (5.37), ɮɭɧɤɰɢɹ Ƚɪɢɧɚ ɪɚɜɧɚ ɩɨ ɩɨɪɹɞɤɭ ɜɟ-
ିଵ
ɥɢɱɢɧɵ ߬ ɜ ɫɥɭɱɚɟ ȁ߱ȁǡหߦ
ɧɟɪɚɜɟɧɫɬɜ. ɍɫɥɨɜɢɟ หߦ
ห߬
p
ห߬
p
ɨɤɪɟɫɬɧɨɫɬɢ ɫɮɟɪɵ Ɏɟɪɦɢ ɜ ɫɥɨɟ ɬɨɥɳɢɧɨɣ ɩɨɪɹɞɤɚ
ɢ ɛɵɫɬɪɨ ɫɬɚɧɨɜɢɬɫɹ ɦɚɥɨɣ ɩɪɢ ɧɚɪɭɲɟɧɢɢ ɷɬɢɯ
ିଵ
ɬɪɟɛɭɟɬ, ɱɬɨɛɵ ɢɦɩɭɥɶɫɧɵɣ ɚɪɝɭɦɟɧɬ p ɥɟɠɚɥ ɜ
ିଵ
ሻ
ሺ
߬
. Ɍɟɩɟɪɶ ɩɨɫɦɨɬɪɢɦ
ݒ
ɧɚ ɜɵɪɚɠɟɧɢɹ (5.12) ɢ (5.13). ȼɫɟ ɮɭɧɤɰɢɢ Ƚɪɢɧɚ ɩɨ ɩɨɪɹɞɤɭ ɜɟɥɢɱɢɧɵ ɜɧɨɫɹɬ
ɨɞɢɧɚɤɨɜɵɣ ɜɤɥɚɞ ɩɨɪɹɞɤɚ ߬ ɢ ɩɨɷɬɨɦɭ ɨɧ ɨɞɢɧɚɤɨɜ ɜ ɨɛɟɢɯ ɞɢɚɝɪɚɦɦɚɯ, ɬɨ ɠɟ
ଶ
ȁ
ɦɨɠɧɨ ɫɤɚɡɚɬɶ ɢ ɩɪɨ ɜɤɥɚɞ ɨɬ ɩɪɢɦɟɫɧɵɯ ɥɢɧɢɣ
ȁ
ݑ
.
Ɉɞɧɚɤɨ ɜ ɩɟɪɜɨɦ ɜɵɪɚɠɟɧɢɢ (ɞɢɚɝɪɚɦɦɚ ɛɟɡ ɩɟɪɟɫɟɱɟɧɢɣ ɩɪɢɦɟɫɧɵɯ ɥɢɧɢɣ)
ɢɧɬɟɝɪɢɪɨɜɚɧɢɟ ɩɨ ɢɦɩɭɥɶɫɚɦ p
ɡɚɦɟɬɚɸɬ ɨɛɴɟɦ ɩɨɪɹɞɤɚ
ɢɦɟɟɬ ɚɪɝɭɦɟɧɬ p
ଵpଶ
ସ
െp, ɢ ɷɬɨɬ ɚɪɝɭɦɟɧɬ ɞɨɥɠɟɧ ɥɟɠɚɬɶ ɜ ɨɤɪɟɫɬɧɨɫɬɢ
ɮɟɪɦɢ-ɫɮɟɪɵ, ɩɪɢ ɭɫɥɨɜɢɢ ɱɬɨ ɜɫɟ ɬɪɢ ɦɨɞɭɥɹ หp
ɩɨɪɹɞɤɚ
. ɗɬɨ ɜɨɡɦɨɠɧɨ ɬɨɥɶɤɨ ɟɫɥɢ p
ɤɨɦ ɦɨɠɟɬ ɩɪɨɛɟɝɚɬɶ ɜɫɸ ɮɟɪɦɢ-ɫɮɟɪɭ, ɨɞɧɚɤɨ ɧɚɪɭɲɚɬɶ ɪɚɜɟɧɫɬɜɨ หp
െpห
ɝɪɚɦɦɟ ɡɚɦɟɬɚɸɬ ɨɛɴɟɦ
ሺ
ɦɨɠɧɨ ɬɨɥɶɤɨ ɧɚ ɜɟɥɢɱɢɧɭ
ିଶ
ሻ
߬
ݒ
. Ɍɨɝɞɚ ɭɫɥɨɜɢɟ ɨɬɧɨɫɢɬɟɥɶɧɨɣ ɦɚɥɨɫɬɢ ɞɢɚɝɪɚɦɦ ɫ ɩɟɪɟɫɟɤɚɸɳɢɦɢɫɹ
ଶ
ɢ pଶ ɢɞɟɬ ɧɟɡɚɜɢɫɢɦɨ, ɢ ɜ ɢɬɨɝɟ ɷɬɢ ɢɦɩɭɥɶɫɵ
ଵ
ିଶ
ሺ
ሻ
ݒ
߬
. ȼɨ ɜɬɨɪɨɣ ɞɢɚɝɪɚɦɦɟ ɨɞɧɚ ɢɡ ɮɭɧɤɰɢɣ Ƚɪɢɧɚ
ȁpȁ
ห, หpଶห,
ଵ
ଵൎpଶ
ሺ
ݒ߬
ିସ
ሺ
ሻ
ݒ
߬
. Ɉɬɧɨɲɟɧɢɟ ɜɬɨɪɨɝɨ ɨɛɴɟɦɚ ɤ ɩɟɪɜɨɦɭ ɟɫɬɶ
ൎെp. ȼɫɹ ɷɬɨ ɤɨɦɛɢɧɚɰɢɹ ɰɟɥɢ-
ିଵ
ሻ
. ȼ ɢɬɨɝɟ ɢɦɩɭɥɶɫɵ ɜɨ ɜɬɨɪɨɣ ɞɢɚ-
ɬɨɠɟ ɞɨɥɠɧɵ ɛɵɬɶ
ଵpଶ
െ
ɩɪɢɦɟɫɧɵɦɢ ɥɢɧɢɹɦɢ ɟɫɬɶ
ݒ
ିଶ
ሻ
߬
اͳǤ
ሺ
ͷǤ͵ͺ
ሺ
ɗɬɨ ɧɟɪɚɜɟɧɫɬɜɨ ɩɨɡɜɨɥɹɟɬ ɪɚɫɫɦɚɬɪɢɜɚɬɶ ɞɜɢɠɟɧɢɟ ɷɥɟɤɬɪɨɧɨɜ ɦɟɠɞɭ
ɫɬɨɥɤɧɨɜɟɧɢɹɦɢ ɫ ɩɪɢɦɟɫɹɦɢ ɤɥɚɫɫɢɱɟɫɤɢ, ɬɚɤ ɤɚɤ ɷɬɨ ɟɫɬɶ ɨɬɧɨɲɟɧɢɟ ɯɚɪɚɤɬɟɪɧɨɣ ɞɥɢɧɵ ɜɨɥɧɵ Ⱦɟ Ȼɪɨɣɥɹ ɷɥɟɤɬɪɨɧɚ ɤ ɞɥɢɧɚ ɫɜɨɛɨɞɧɨɝɨ ɩɪɨɛɟɝɚ. ɉɪɢ ɜɵɩɨɥɧɟɧɢɢ ɭɫɥɨɜɢɹ (5.38) ɩɪɢɦɟɧɢɦɨ ɤɢɧɟɬɢɱɟɫɤɨɟ ɭɪɚɜɧɟɧɢɟ, ɩɨɷɬɨɦɭ ɫɥɟɞɭɟɬ ɨɠɢɞɚɬɶ ɱɬɨ ɩɨɥɭɱɟɧɧɵɟ ɪɟɡɭɥɶɬɚɬɵ ɦɟɬɨɞɨɦ ɮɭɧɤɰɢɣ Ƚɪɢɧɚ ɧɟ ɞɨɥɠɧɵ ɨɬɥɢɱɚɬɶɫɹ
ɨɬ ɪɟɡɭɥɶɬɚɬɨɜ, ɩɨɥɭɱɟɧɧɵɯ ɢɡ ɤɢɧɟɬɢɱɟɫɤɨɝɨ ɭɪɚɜɧɟɧɢɹ, ɜ ɫɥɭɱɚɟ ɩɪɟɧɟɛɪɟɠɟɧɢɹ ɞɢɚɝɪɚɦɦ ɫ ɩɟɪɟɫɟɱɟɧɢɟɦ ɩɪɢɦɟɫɧɵɯ ɥɢɧɢɣ.
Ʉɫɬɚɬɢ ɨɬɦɟɬɢɦ, ɱɬɨ ɜ ɞɜɭɦɟɪɧɨɦ ɫɥɭɱɚɟ ɦɚɥɨɫɬɶ ɞɢɚɝɪɚɦɦ ɫ ɩɟɪɟɫɟɱɟɧɢɟɦ
ሺ
ɛɭɞɟɬ
ݒ
ିଵ
ሻ
߬
, ɚ ɜ ɨɞɧɨɦɟɪɧɨɦ ɦɚɥɨɫɬɶ ɨɬɫɭɬɫɬɜɭɟɬ ɜɨɜɫɟ, ɢ ɜɫɟ ɞɢɚɝɪɚɦɦɵ
ɧɟɨɛɯɨɞɢɦɨ ɭɱɢɬɵɜɚɬɶ. ɗɬɨ ɨɡɧɚɱɚɟɬ, ɱɬɨ ɞɥɹ ɨɞɧɨɦɟɪɧɨɝɨ ɷɥɟɤɬɪɨɧɧɨɝɨ ɝɚɡɚ
ɤɢɧɟɬɢɱɟɫɤɨɣ ɩɨɞɯɨɞ ɧɟɩɪɢɦɟɧɢɦ.
5.2. ɍɫɪɟɞɧɟɧɢɟ ɩɪɨɜɨɞɢɦɨɫɬɢ.
Ɏɨɪɦɭɥɚ Ⱦɪɭɞɟ
ȼ ɷɬɨɣ ɝɥɚɜɟ ɦɵ ɪɚɫɫɦɨɬɪɢɦ ɬɟɯɧɢɤɭ ɭɫɪɟɞɧɟɧɢɹ ɩɨ ɛɟɫɩɨɪɹɞɤɭ ɩɪɨɜɨɞɢɦɨɫɬɢ ɫɢɫɬɟɦɵ ɜɵɪɨɠɞɟɧɧɵɯ ɷɥɟɤɬɪɨɧɨɜ. ɉɪɨɜɨɞɢɦɨɫɬɶ, ɷɬɨ ɱɚɫɬɧɵɣ ɫɥɭɱɚɣ ɞɢɧɚɦɢɱɟɫɤɨɣ (ɨɛɨɛɳɟɧɧɨɣ) ɜɨɫɩɪɢɢɦɱɢɜɨɫɬɢ, ɧɨ ɩɪɨɞɟɦɨɧɫɬɪɢɪɨɜɚɧɧɵɣ ɩɨɞɯɨɞ
ɛɟɡ ɫɭɳɟɫɬɜɟɧɧɵɯ ɢɡɦɟɧɟɧɢɣ ɩɪɢɦɟɧɢɦ ɢ ɤ ɡɚɞɚɱɟ ɭɫɪɟɞɧɟɧɢɹ ɞɪɭɝɢɯ ɮɭɧɤɰɢɣ
ɨɬɤɥɢɤɚ.
ሻ
50
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