DEGREE. 
sl determined in two different places, as is ufually done, 
atthe two extremities of the chain, we fhall have a etd 
file ethod of verification, fince the calculated and o 
ved azimuths ou zree 
Te muft finally be abferved, Cant the point X has rather 
greater latitude than the point L. Let a be the latitude of 
the point L, v the radius of curvature of the meridian towards 
L, y the dittance LX, R the number of-feconds, comprifed 
in the radius of the tables, we fhall find that the latitude of 
2 
a ry e Y oF 
the point X isa +43R (=) tang. A, where the correction 
will be expretfed in feconds. 
Though the different portions of a meridian line may be 
ealculated by either of thefe methods, yet the problem in its 
tothe th : is required to determ 
the latitudes of all t ations, their exits with regard 
to each Te and hele i fference of longitu m the point 
the diftance between the Ty of aay two fta- 
lafly he arc of the meridian intercepted by the ex- 
treme ftations. When the triangles are large, and diverge con- 
n to be cor- 
figure of the earth ; that is u 
confidered not as {pherical ba {pheroidical. very ufeful 
theorem is given in the Englifh furvey, for the calculation of 
we fhail extra & from Puiffant’s * Traité 
e formule, which form the 
bafis of practical rules given si Delambre and Legendre. 
The principal obje&t of thefe formule is to obtain alge- 
braical expreflions for computing the value of the following 
quantities: 
The radius of a circle parallel to the equator in any lati- 
tude ;—the normal, or radius of curvature of a great circle 
or 
m s extremities ;— 
compreffion, or ellipticity the terreftrial {pheroid, deduced 
from a meafure of two arcs ;—the eccentricity ;—the length 
of a ftandard meafure as ne French metre, which is affumed 
equal to the ysonecd"" part of the meridian. 
Inveftigation of Formule for expreffin ing in Terms of th 
tude diferent . Parts oe 2 Meridian, the Earth being ‘hapued 
an ellipfoid of Revolu 
Let CE be radius oe the eanator (fig. O4.): P the 
ole. 
. If from the . A atangent AT be drawn to the ellip- 
tic are BAE; MA will be the normal to the point A, and 
ZALT= PATS latitude of the point A. 
The equation to the ellinte is a? y? + B? x? = a? 3; and 
for the point A, whofe co-ordinates are x ‘59’, we have a? y” 
+ Ba? =a BD. At — a point A the equation to the 
')3 if y= 0, thenCL 
normal ALis y — jy’ = = 
orx = a x’; from which it is _ to conclude that 
a 
—_ e 12 
ae aah 
n fin, L; and confes 
the normal AL 
Let ALF = 
es 
— 
—y 
L, then »’ 
quently y? = [e+ aoe | tine L; hence y* 
Gn L. 
a= (a — 2) fix Lats (1). 
Anda = — ae | 
a 
If in this equation we put a = 1 and - and 8’ we fhall 
a — 6 : 
have so =e; or 1 — 47 = ce’, ¢ exprefling the 
aaah, then n= (1 —e) (1 — @ fin? LY 
moe in’ Ly’ °3) 
And ia equation (1 1) will become AF = 9! = 
(1 — ¢) fin. L 
I— . i L)} (4). 
n the Hee peal ass vi bales of the ellipfe is 
dias into y? = (1 oe and by the preced-« 
co 
to 7° so Ly 
ree : the value of the radius of a parallel of latitude to 
the p 
‘a ie 
may be cee into é& x’; but x 
ine teeeees 
3)3 eresore = —= hn Ly 
All the values a Sone are relative to the greater 
axis taken as the li 
The fae mode of eAcilation will lead us to the values of 
AM and , &e. 
oo x an B 
a 
ing equation CF = x! = 
e manner, the value of CL, found as above, 
CL= ‘is given in equa- 
(6) * 
= a’ 3, and let the values of x be 
now taken on the ler axis. Let the normal A M= 7’, 
we fhall have for the point A; y? = n” cof. L; but if in 
equation (2) a be ae Sa into 2 and vice verfa, and fine 
for cofine AM = n’ = aa ) cof.? L)4, 
and fince 6 = 4’ when a = 1, 
I 
oe 
a 
a 
‘ e 3 
Gey La + a = fine L) | 
I 
(i —é fin L)¥ (7). 
Since y? n? Col? 1a. 9 
col,? L 
1-—eé fn’ L 
(1 —e?) — x 
oy 
2 
By the equation of the ellipfe ¥’ 
I— ¢ 
from has we o, deduce 
— 2) fi 
2 emne 
oe ce ra 
As we have found above CL = FE x's we 
fhall in the prefent cafe have CMT = - —— x = 
— 
~ 35 fubstituting for x its Vea a the preceding equas 
con, we obtain CM = 
fin. L. (8), 
As to the value of AC, it is evidently reprefented by: 
WV x? + 9%, fo that whether we take the above values of 
; 6 ay 
(ie iby = 
