DEG 
Bet deol? £ o cof.3 Z). 
ait cof, 3 gcof. Z (1+tang. 39 fin. Z tang. Z tang. 
“pe ne £ofin.*Z tang.’ L + $fin? £¢col.*Z). 
— (9 cof. Z +4 ¢ fin.@ fin.? Z tang. L) 
Such is the approximate value of the > latitude of the point B, 
. Now to take into confideration the eccentricity, we mult re- 
colle& that the exact latitude of B = go? —PCB; but fince 
CB=PMB—NBM,andPMP =90° — L’—(L—dL 
it follows that the exa& latitude B= L—dL 
~~ 
Subftitute for d L its a found above, and font its 
value as alfo above determ 
Then cae! the enact aa ude by L’, we fhall have 
L! = cof, Z a 
L - Re iia. a fin. 
In this formula the ne m 3 ey &c. may be neglected ; then 
the faGtor of dL will be reduced to(t + ecof?L). As to 
the quantity @, it fhould be exprefled in (oonae and it has 
already been fhewn if K reprefent the chord of an arc, % 
re ohn Z tang. ore + col 
will be equal to ar G 
Proceeding to find the az zimuth Z’, or that of the point A 
on the horizon of B, the triangle P A B gives this equation : 
Tang. §(B4A cot. 4 Pcof.$(P B— PA) 
ang. 2 (B+ A) = cof Z FEB + PA) 
cot. 4 P cof % Z(L— L’) 
cot. (80° — a ds) 
_ cot. % P cof. 3 (L —L’) 
fin. 3(L + L’) 
tang. 
Cot. £(B + A) = tang. (go? — 
tang. = Ate). 
and fince cot. = 
3(B + A))= 
Asi e —i ae a A) and Pare always Het {mall angles : 
3 Pfia. $(L +L’). 
—L) ° 
P fin. at 
? 
and, B=(180—A) 
col 4 
Ly) 
This formula gives the i dion BA mond al 3 if 
it be fouth of the weft, we muft add 180°, and then 
) 
, 7 P fin. 3(L 
ore cot. (= LF 
The fame triangle gives 
fn. P= o% -ABfn.A_ "fin. @ fin. Z es 
” cof. L ee L $y 
From whence sor be deduc Z 
3 n,Q fin Z a L4+UL' 
Zane Te cal ELL) oot TL 
2 aes @ fin. iGhite) 
ae eo a , 
i t 
= ake ? 7 Z = ead idL+fin.3dh = 
cof. a 
= — > tin. Z tang. L’ — 4d 2! 
for d L “oti its wide (2), we fhall hee (seeing the 
terms of the t order) fince 2 fin. Z cof. Z = fin. 2 Z, 
Z' = ¥80 + Z - ofin. tang. es (fina Z 
This is the azimuth of A, fee the horizon of B, to 
which there is nothing to add for the aesany or the earth, 
the effe& of which is infenfible. 
As to the longitude M’ of the point B, it is clear that 
it refults from equation (3); for if M = longi itude of A; 
Ul 
— M. 
R E £E. 
And fin. P, or fin. (M’—M) = ©: | 
or nearly exaét M’ = M +- : ae 3 ‘ 
If in this expreffion for L’ we fubfitute its approximat 
value L — @ cof. Z 
@ fin, Z 
M=M 
F ton(L—oeal.Z) 
Expanding the dogg nal r, aud con fi aie gcof. Z asa 
{mall arc, 
@fn Z fin Z 
M’=M a 
= cof, L (1 4+ Qcol. Ztang. L Tees cof, £ 
(1 — %cof. Z tang. L). 
fn. Z Ls 
And, finally, M’=M + ¢ — - to fn. 2Z— ws, 
To ili age the preceding refults ; 
let p = radius of the equator, 
: = eeentaeiy, 
@ = the arc exprefled 1 in fecords, correfponding with the 
chord K of a terreflrial arc, which is one Ede 
of the triangle 
L = the latitude known of one extremity of the chord K, 
L’ = the latitude fought of the other, 
M = the longitude kn. ea a reckoned from the fouth to 
M’ = the longitude ae the weft, from o to 369, 
== the azimuth kno i aned guecoane 
Z' = azimuth ioaghe pees ° 
= ia p(t — eth L). (a) 
L’ = L— (gcof.Z + £¢ fin, O fin? Z tang. L)(r + 2? 
~ cof? L). 
Z' = 180+ Z— olin. Z tang. L’— O fin. ¢ fis. oO 
M=M+¢ * A = a tt 2 borin? Ze (d) 
dL, or difference of the parallels in terms of the “landard 
meafure, as toifes, fathoms, or metres. 
K cof. £ @cof. Z) + (Kcof.4 cof. Z) (tang. 
I @ fin. Z tang. Z. tang ane (K cof. “V9 cof. Z) (fin. 
49 tang. £ fin.* L tang? L) + 
(fum of the three firlt ee (1 — é fin. L) 
—_— 
6 6 
The fourth term of Are expreffion is the excefs of the are 
; here the radius is that of the 
in 
above the fine, or  — 
earth fuppofed fpherical, if for greater exactnefa we cmploy 
the radius of curvature of the arc A.B, or r= cae TDP 
and inflead of re its value found as above, we frail have 
the term in queftion 
Example I. 
Let the rasan of A be equal 48° 50’ 49%.7; log. of 
AB (79095.6 feet) = 4.8981525 = K;Z the ta of 
AB, or angle '. 7 27", evired ‘the latitude of B? 
” KR’ fin. *L KR 
$ p oe se al e(t + fin, 71.) 
Sin.? L - : 7535410 
. = +7.706329 
0.5 - - ze “689700 ; ; 
1.0C0g000 
_? 2220143 39 == 0.0910G4y 
9.99265 = 0.9483051 
Uu 99. oe Brought 
