172 GEODESIC rN'VESTIGATIOyS. 



Problem 9- 



Given the azimuths A', A", and the length s and circular measure 2 of the 

 geodesic arc between the stations ; to find the latitudes I', I", the difference 

 of longitude co, and the other various entities determined in the solution of 

 problem 1. 



We hare (see formtila 36) the ares S^^„ S^J)^, or sides z„ z^, of the 

 triangles S,FD^„ '^.,PD, expressed by— 



r 



z, = sin ^ - ^°^' ^^ ;. — (90° — 12) 



1 + -— — — . sm -frS I 

 L P^? s, - J 



the arc within the brackets being greater than 90^ ; 



f" 

 I 

 z,= (90' + i2)-sin-i^ 



1 •+ _-_ — sm i^2 I 



the arc within the brackets being less than 90"^ 



We know that z^ is a little less than 2, and that z,, differs little from 

 2. But in order to obtain z^ and z., with great accuracy, we require only 

 sufficiently approximate values of S, and R,,. These we can find by assuming, 

 for the occasion, that the arc 10 (which bisects the angle A) cuts the arcs 

 FS^, FS^, at angles respectirely equal to A' and A"; and that the portion of 

 it intercepted between them is equal to 2. Then putting i>^, D^ for the 

 points in which arc 10 cuts FS^ FS^,, we hare — 



ta.i(i^„+^A)= ::;*<j:;j:j .t.ni. 



from which we find colatitudes FD^, P^co that cannot differ in any case from 

 the absolutely rigorous values by more than a few seconds. 

 Then with approximate values of the latitudes obtained from 



r = 90° — FD^ ; I" = 90° — PDoo ; 



we can find values for the lengths of the normals S„ E„ (terminating in the 

 polar axis), which are equally as efiicienfc in determining extremely correct 

 values of z^, z^„ as if they were found with absolute accuracy. 



Now having found the values of z, and «,„ we can find the angles D,, D^, 

 by means of formulae (21) which give — 



. ^ cos *2 . 



sm !>,— J—— . sm A 



cos (z,, — *2) 



. cos i2 . „ 



sm !>,,= —- f— — . 6in A" 



cos {z, — i2) 



Then in the triangles S^FF>^^, S^,PD„ we have the bases r„ z^^, and tlie angles 

 at the bases, so that we can find their common vertical angle w, and their 

 sides l^, I^, which are the complements of the latitudes — 



+ i c 7- ^ 7 \ coBi (A' — D„) 



tan i {L„ 4- Q = ^^3^(^,^.jy tan k z. 



