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Table 1 Cosines in Terms of Sines(Copied from Proc. Phys. Soc, xxxii, opposite p. 258)

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sin α = sin θ + g R , sin α = n n sin α , θ = θ + α α , g = cos α + cos θ cos α + cos θ g .
g g = sin α sin θ sin α sin θ = sin α θ 2 cos α + θ 2 sin α θ 2 cos α + θ 2 = cos 2 α + θ α 2 cos α + θ 2 ,
g g = cos 2 α + cos ( θ α ) cos ( α + α ) + cos ( θ α ) = cos ( α α ) + cos ( α + θ ) 1 + cos ( α + θ ) ;
g g = cos 2 α + cos ( θ α ) + cos ( α α ) + cos ( α + θ ) 1 + cos ( α + θ ) + cos ( α + α ) + cos ( θ α ) = 1 + cos ( α + α ) + cos ( θ α ) + cos ( θ + α ) + 2 ( sin α sin α ) ( sin α + sin θ ) 1 + cos ( α + α ) + cos ( θ α ) + cos ( α + θ ) = 1 + 2 ( sin α sin α ) ( sin α + sin θ ) 1 + cos ( α + α ) + cos ( θ α ) + cos ( α + θ ) .
g = g + 4 g ( sin α sin α ) ( sin α + sin θ ) ( sin θ sin α + sin α ) 2 ( cos θ + cos α + sin α ) 2 1
sin α k = sin θ k + g k R k , n k + 1 sin α k = n k sin α k , g k = g k + 4 g k ( sin α k sin α k ) ( sin α k + sin θ k ) ( sin θ k sin α k + sin α k ) 2 + ( cos θ k + cos α k + cos α k ) 2 1 sin θ k + 1 = sin α k g k R k , g k + 1 g k + d k sin θ k + 1 ( where d k = A k A k + 1 )
tan θ 1 = h 1 υ 1 r 1 + r 1 2 h 1 2 , g 1 = υ 1 sin θ 1 .
υ 3 = + 092636 inch .
p = n n p , sin α = p R , sin α = p R , θ = θ + α α ;
p k + 1 = p k + a k sin k + 1 ,
a k = d k + r k + 1 r k .
υ 3 = υ 2 d 2 , sin θ 4 = n 3 sin θ 3 / n 4 , υ 3 = υ 3 tan θ 3 / tan θ 4 .
n c l = n c l .
x : x = l : l ,
n c x = n c x .
n x = n x + n n e .
n u = n u + n n r .
EC : BC = sin / EB _ C : sin / CG _ B ,
/ EB _ C = π ( α + α ) 2 and / CG _ B = π + ( θ + θ ) 2 ,
e = cos α + α 2 cos θ + θ 2 r .
n c n c = n n r cos θ + θ 2 cos α + α 2 .