124 REPORT ON 



We have also 



X=i7cos£ F= — II sin£ Z= II tan 



X'= H' cos I ' Y'= — H sin J ' 



Substituting these values in equations (1), (2), and (3), and dividing by H, we 

 have 



H' P 



— cos C, ' = (1 + a) cos £ — & sin £ -(- c tan + — (4) 



— ~sm%' = d cos f-(l+e) sin £ -j-/ tan + 5* (5) 



|! = flr cos £ - ft sin £ + (1 +7.) tan + * (6) 



Equation (6) may be written 



n t Z' . cos £ 7 sin t , , . -R , a , 



From equations (4) and (5) we obtain the following: 



(4) cos £ — (5) sin £ gives after some reductions 



"msi 1 \ a ~\~ e \ (n ten fi \ P \ n na ? I Z+o-n A I Q 



II 



cos 5 = 1 4- °L+ e + (c tan +±1) cos £ — (/tan _|_ j) sin £ 



^cos2£ — cZ Jl^sin2£ 



(4) sin £ -|- (5) cos £ gives after some reductions 

 J sin S = ^ + (c tan 0+ J) sin£ + (/tan +-|) cos £ 



+ a ~ e sin2^ + ^4 1 ^ cos2 ^ ( 8 ) 



Now let 



ctan0+J = X23 /tan0 + ^ = ^S 



Then from equations (7) and (8) we get the following : 



JL cos 5 = 1 + 23 cos £ — g sin £ + © cos 2£ — @ sin 2£ (9) 



j^' sin 8 = % + 23 sin £ + g cos £ + 2) sin 2£ -+- @ cos 2£ (10) 



Dividing (10) by (9), 



s _ 21 + 33 sin £ -f g cos £ -f £) sin 2 £ + 6 cos 2 £ m) 



l+23cos£ — gsin£ + £)cos2£ — gain 2 f ^ 



