The rule “to ‘swing’ the azimuths to the adjacent quadrants” as stated on page 485, and to reverse the columns of the traverse tables as required in the last paragraph of page 486, may be a “ rule of thumb ” to obtain the result, but it does not show the “ reason why.”
From the figure given above, the deductions of the same formulae are clearer and simpler and at the same time the reasons for the operations are apparent.
After having found the azimuths and then the directions (α1 and α2) of the altitude differences, and their values (h1 and h2) ; and after having plotted them and their lines of position—draw through the assumed point (B), the lines BM and BN, parallel to the lines of position—thus forming the parallelogram BMZN.
The quadralateral BCZE is formed by the altitude differences (h1 and h2) and their lines of position.
In the quadralateral BCZE
angle CBE = difference in direction of h2 and h1
= α2— α1 = Α
angle CZE = 180°— Α
In the parallelogram BMZN
angle MZN = 180 — Α and
angle BMZ = BNZ = Α
side BM = h1 cosec
side MZ = side BN = h2 cosec Α
direction of BM = α2 — 90°
direction of MZ = α1+ 90°
Difference in Lat. between B and Z
= BK + KD
= BM cos (α2 — 90) + MZ cos (α1 + 90)
= h1 cosec Α sin α2 + h2 cosec Α sin α1
D. Lat. = cosec Α (h1 sin α2 + h1 sin α1)
Departure between B and Z
= MK — ML = KL = DZ
= BM sin(α2 — 90) — MZ sin(α1 + 90)
= h1 cosec Α cos α2 — h2 cosec Α cos α1
Dep. =cosec A (h1 cos α2 — h2. cos α1)
From the figure it is seen that the point of intersection of the two lines of position may be found by working the traverse for the distance BM ( = h1 cosec A) on course (α2 — 90), and distance MZ ( = h1 cosec A) on course (α1 + 90).
Following the above method, the directions of the factors (h cosec A =9.5 and h cosec A = 7.4) in the solution of the problem on page 6 of the accompanying paper would be S. 48° E. and S. 60° W. instead of those given, and the columns of the traverse tables would be used in the usual manner.
Similar changes would be made in the solutions of the other problems, but both methods give the same results.