27.2 Calculation of the Integral (27.5)
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which designates
1 (α, α 0 , ω) = sin α cos α cos α 0 + H tg ω,
2 (α, α 0 , ω) = sin α cos α 0 tg ω+
+ cos α
sin
2 α 0
cos 2 α + tg 2 ω
− sin
2 α cos 2 α 0 ,
whereas
H = H (α, α 0 , ω) =
sin
2 α 0
1 + tg 2 ω
− sin
2 α.
Taking into account the latter formulas and expressions (20.9), let us calculate
η λ cos ω 0 + ξ λ sin ω 0 =
cos ω
sin α 0
(cos ω 0 sin α + H sin ω 0 ).
(27.7)
Now, by calculating the product of single vectors #» n and #» ν , and by attracting the
expressions (27.6), after some identical conversions, we will find
cos w 0 =
cos α cos α 0
1 + tg
2 ω
+ H sin α tg ω
cos 2 α + tg 2 ω
.
(27.8)
Let us express the differential dw 0 via dα 0 . To do it, let us use the equation
sin w 0
#»
λ = #» n × #» ν .
Hence we have
sin w 0 =
sin α cos α 0 tg ω − H cos α
cos ω
cos 2 α + tg 2 ω
.
By differentiating the latter formula, we obtain
cos w 0 dw 0 = −
H sin α sin αα 0 tg ω + T sin α 0
H cos ω
cos 2 α + tg 2 ω
dα 0 ,
(27.9)
where
T = T (α, α 0 , ω) = cos α cos α 0
1 + tg
2 ω
.
Based on the obtained ratios (27.7)–(27.9), formula (27.5) is converted into