GEODESIC INYESTIGATIONS. 157 



From these we have — 



cos V sin w , ,„^ 



sm«, = -_ ^ ; ^v i (27) 



J (sin ^" — tan \ 2' cos Z' sin co)- + (cos ?' sin co)- .> 



cos Z" sin 03 

 sin z,, = J— -_ ^ . i (28) 



< (sin A' — tan ^ 2" cos Z" sin co)'^ + (cos I" sin co)^ V 



And from these and equations (22) we have — ■ 



sin X'„= \ (sin^" — tan i 2' cos V sin w)~ + (cos V sin w)- i (29) 



r . ; . •) * 



sin D^ = \ (sin A' — tan | 2" cos Z" sin co)^ + (cos I" sin co)- > (30) 



© 



"We can express z„ z^„ in other useful forms. 



In the plane triangle S'S"Z, we know that the side S'Z^ is S^ ; that S'S" 

 is /t ; and that the angle S"S'Z^ = 90° — ^ 2'. 

 From it we have — 



S"Z, — (F + S/ — 2.JC.B, sin i 2')* 

 k. cos i 2' 



Bin ^, = 



S"Z, 



Ic cos i 2' 



sm z, ■=. 



) TA -1. 7? 2 O I- J? cJn i ■^' . 



ih- -\- B;- — 21cB, sin i 2' 1 



-, . 7 2 s . sin i 2, 

 and, since /fc ::= 



sm z, =: 



2 



s . sin 2 



I (-R, 2)2 — 45. sin ^ ^ 2 (i?, 2 — 5) | 

 Similarly, 



7(7. cos i 2" , 



(31) 



'"■"■ "11 7 N 2 



I (F + :RJ — 2.^.i?„ sin i 2" J ^ 



s-sin 2 , ■ ( (32) 



sm z„ — — \ \ 



\ {E„ 2)2 — 4.5. 8in2 i 2 (i?,, 2 — s) J ^ 



If we put p for the mean radius of curvature of the arc s ; then 2 = _. 



and sin^ ^ 2 r^i ( — I nearly when s is not more than 1°. Hence, for such 

 small arcs, we have — 



""'' ' . ... (33) 



Bin 2 



{(^y-\{~y- i^)] 



