344 Mr. Ivory on the Properties of a Line of shortest 
And, by passing from the expression of the sine to that of the 
arc itself, I have found, 
z=yWV1+e?cos*l + — sin°y. (E) 
If we compare this expression with the formula (D’) we shall 
readily deduce 
v=yt = ~—sin ? as (F) 
Hence, in measurements to a certain extent, x and y may be 
regarded as equal ; and either of them will give the amplitude 
of the length measured. It is easy to substitute y for 2 in the 
expression of the geodetical line already given. 
It is requisite to observe that in low latitudes the value of 
. t PN - e,° 
sin y (= =a) is the quotient of two small quantities; and 
that an error in w will be greatly augmented in y. It is there- 
fore only in considerable latitudes that x can be safely deter- 
mined by means of the azimuth. ‘ 
The foregoing analysis will enable us to deduce the diffe- 
rence of longitude directly from the variation in azimuth. We 
have already found, 
: sin 7 sin z 1 r 
RUN SC a ant cinka 7 ae ee 
therefore assume 
sin w A/1 +e? costl f 
ae Tae ge eee 
sin § = : 
sin / 
then sin 6 == ES 
a/ cos?i+ sin 7/ sin? x” ’ 
tan z * 
—— = tan 6. 
cos 1 
But we have likewise found 
ay an tan x , @ 3es 3 } . 
9 = arc tan (* ) cos 1 } = 5 cas* lt x<ez: 
wherefore, we get, 
” _ sinw afl +e%cos?l 
sin (= SCRE aay DRT aER > 
g=h— § = cos — 3 cos */ i x arc tan.(tan 6 cos /). 
This formula is already very exact, and will extend to an am- 
plitude of 10° or 12° from the beginning of the geodetical 
line: but the method we have employed may be carried to 
any required degree of approximation. 
In the Conn. des Tems 1828, I find an example very pro- 
per to illustrate the foregoing calculations. Ina perpendicu- 
lar to the meridian commencing in latitude 45° M. Puissant 
has 
