DEG 
x", y°, or have recourfe to 7. eq ations San (5), we have 
fin 
ll ee 5) : ‘ 
a cs = eho (9) 
This formula is fufceptible of a more commodious form: 
Imagitie a {phere circumf{cribiag the ellipfoid, which has 
for its radius that of the equator, the angle Ca E = FaT 
= ill be the latitude of the point a on th e fphere; but 
a points A, : Ha the a abfeifle C F; therefore if we 
eAF= a the bee of the circle 
and ellipfe a be en, ty” 
2— £" ve. 
I x? 
= (1 — x’) 
eliminating x? ; 
a F is the fine of 4 face aC = 1, therefore fine °, 
gee 
andthe equation (9)becomes AC = 
I 
(1 — é fin. 10). 
Now to find ie ae of A, divide in the preceding equa- 
= b. 
tions of the two curves one by the other, and z 
J 
/ 
By infpeGion of the figure it will be feen that oF 
y! 
y” 
y” I 
—— = —~—-—— therefore = 
FT tang. A 
From thefe two values of, - refults that tang. A = @! 
tang. L, 
If B betaken oe that cof. B= d’,then1 —b’=1 — fin. 
in.? 3B, tang. A = cof. B tang. L, and 2 
2 fins? SB=1+4+8, therefore tang.” 3B 
(22 —_2— 
| 
» and by the expanfion of a_trigonometrical 
— A = tang.? 
+ Jtang.°2 B fin. "6 L, 
1—J8 I 
a0) 2L— 4 (— 
Les (25) foo, 
This feries is rendered very converging by ae: the 
and 
L—as TF 
ratio of the axis of the ellipfe m:2n, for d = 
— J! am 
ee oe m, ngenerally differ but one 
i+ m m+n n ma 
unit from each other: thus LL — a= (- : - | fn. 2 L 
) ?finn 4 Le + 4G (- : : 3 fin. &-L, fince 
The firft term of this feries is 
I 
m+n 
= 334,” = 333 nearly. 
fafficien: 
rh to find A C; in the exprefflion of its a (1 — é 
— oe for a its equal as found abov 
od an expreffion for 
the meridian, j it fhould be re nembered 
fecond order it is always equal to the 
divided by one quarter the iquare of the parameter, 
Therefore for tne lat. L 
_ 2° (i me)? ( — # fin. ?L) 43. 
+P 2 
but ip? = ape » hence R = (1 — ee 
fin? L)3, (11). 
£B fin. 2 L — $tang.* 2B fin. 
&c. 
* fin, 
the radius of clare of 
RE E, 
If this value be required in terms of the ellipticity of the 
earth, the longer or equatorial axis being taken as unity, let 
a—&b 
= w#, and becaufe r — 3? = eandd =i —« 
a 
=I—(i~a)?=2e 
hence R = ee a+?) (1 —{2%— — 2) fin? L)~2. 
Expanding this negative ala and rejecting the fecond 
powers and thofe ian ye hav 
= — 3 oe *L). (12). 
We may oe now to the rectification of the curve or 
arc - the sapdb 
c between the equator and the point whofe 
1. 
ts 
latitade | is Li its differentia 
Vd x! x? dy? =dx' / 4a (vid. La~ 
cof. L L 
7 ; : fe —_—_ -——_——— Ten a 5°) 
croix cal. diff.) Since x” = Ca 
dcof, L , co Ld Eas L) 
2 eta L)e 
- nee ; — 
cof.? L 
Ti =e fine L)3 
243 
alles BU ramen TV a 
L 
But when x! increafes, 
dx! = —dL fin. L( 
) 
(1 Finis * 
Gi mar 
= Pin? L 
a 
Tn the expreffion - d8, ee x! “ad d x’, fubftitute their 
values as above ; 
dS= meds ( — é fin? L)?@¢d@L-, and by the: ~ 
binomiai theorem. 
1+ 3éfneL i. 
Se4 fin, 4L 4 3.3.2 68 
I—2 
fin. ©L . dl; then, by changing the powers of the 
fines and cofines of the multiple arcs (vid. La Croix’s Diff. 
Cal. N° 199.) 
dS _§ 3 ea 2% Saat ae 
ae car’ F274 eet 
C58 ey bad 
L223 52 P 
3 1 a 3-5 4 3+5:7, 665 | 
—{3 ed a0 S. ~6 14.2523 
od det ah 
oes 2 Se5 27 6 kh ry 
+13 3 2.4.6 ae} dL cof. 
4L 
oe dLcof.6 L: 
pt. 25° v i “° q 
Or, for concifenefs, 
Ga =mdUL—adLeol.2L+pdLeof.4L “— 
I—é 
Leof.6 L; and by integration, 
ap Se LG afin. 2L+ipfn.4gL— 
+ (33). 
Here the integral is complete, fince the arc S vanifhing at 
the ae shine » the conftant quantity neceffanly a 
comes n 
It is aa that for another arc S, terminating in the la- 
tude L! 
tit . 
tqgfn6L 
PS 
