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2
‚θ2
(B14)
Adding (B13) and (B14) we get

L2 f + L2 f
13 23



© 2001 by Chapman & Hall/CRC
‚2f ‚2f
‚f
2
= ’ cot θ2 2 + cot θ2 +2
‚θ2
‚θ1 ‚θ2
where we have used (B10) and (B11).
Hence from (B9)

‚2f ‚2f ‚2f
‚f
L2 f 2
=’ 2 + cot θ2 ‚θ 2 + cot θ2 ‚θ + ‚θ 2
2
‚θ1 2
1 2


‚2f ‚2f
‚f
2
= ’ cosec θ2 2 + cot θ2 +2
‚θ2
‚θ1 ‚θ2

1 ‚2f ‚2f
1 ‚f
=’ + cos θ2 + sin θ2 2
sin2 θ2 ‚θ1
2 sin θ2 ‚θ2 ‚θ2

1 ‚2f 1 ‚ ‚f
=’ + sin θ2
sin2 θ2 ‚θ1
2 sin θ2 ‚θ2 ‚θ2

L2 f 1 ‚ ‚f
1
= ’’ + sin θ2
sin2 θ2 sin θ2 ‚θ2 ‚θ2
(B15)
In general

L2
1 ‚ ‚
’ k’1
L2 k’1
=’ sin θk (B16)
k
sin2 θk
k’1
θk ‚θk ‚θk
sin




© 2001 by Chapman & Hall/CRC

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