Mr. Burns on finding the Latitudes, &c. 407 
Then, cosa = cosa. cos 6 + cosx#.sina.sind ..... (1) 
cos b = cosa.cosé + cos(x + y).sinaA.sind.. (2) 
cos ¢ = cosA. cos} + cos(e +2).sina.sind.. (3) 
Subtracting (2) from (1), 
cos a — cosh = {cos x — cos (v + y)} .sindA. sind... (4) 
Subtracting (3) from (1), 
cos a — cosc = {cos x — cos (x + z)}. sind. sind 
, cosa — cosh  cosr—cos(r+y) cos — (cosr.cosy — sinz .siny) 
cosa — cose cosr—cos(x+x)  cosxr—(cosx.cosz — sinz.sinz) 
__ l—cosy + tanz.siny 
~~ 1— cos s+ tan z. sing” 
Hence {(cosa@ — cos 6). sinz — (cosa — cosc). sin y} tan x 
= (cosc — cos a). cosy — (cos b — cos a). cos z+ cos b— cose 
sheets 2S (cos c — cosa) cosy — sets b — cosa) cos x + See SE c (a) 
—(cos a — cos c) sin y + (cos a — cos b) sin z 
Now z is the time from noon of the observation nearest it; 
and y, x, the sum and one of the given intervals between the 
observations. Hence the true time is readily determined for 
either observation. 
cos a — cosb 
cos 2 — cos (x TEP and 
from (1), cos A cosé = cos a — sin Asin é cosa. 
Again, we have from (4) sin A sin? = 
cos a — cosh 
Hence, cos A cos § = cosa — cosx (— 
cosa — cos (x+ y) 
*.. sin Asin 8 + cosa cos 6 
cosa — cosb cos a — cos b 
ae cosx — cos (x +y) TePtiks é r — cos a) + 8032 
And, sin A sin 8 — cos A cos 8 
cosa — cosb 
+ cos2. ( 
cosa — cos6 
cos x — cos (ar +y) 
= cos — cos (x + y) Fa Sor the 
cosa — cosh 
Or, cos (A — 8) = (1 ~ cos 2). } ee} +cosa.. (p) 
§ cosa— cosh 
* l cosaw—cos (x+y) ; —cosa.. (y) 
The following example will illustrate the use of these for- 
mula. Suppose at 9" 51™ 58% per watch, the sun’s correct al- 
titude was found 21° 11'; at 10" 48™ 545 it was 24° 40/, and at 
115 29™ 42° it was 26° O'. Required the apparent time, when 
And, cos(A+ @) = (1+ cos z) 
Euler, at page 356 of the Gentleman’s Mathematical Companion for the 
year 1815. We may also refer him to pages 624 and 71 of the same work 
for the years 1816 and 1821] respectively, for various solutions of two ana- 
logous problems.—Eprr. 
the 
