188 A Method for ascertaining Latitude and Time 
‘dh cos A is the same as was obtained by supposing r=-0; 
the term dh sin A tang +a will be a very small trifle. 
As to the term ase Paka 
sin av 
or to very little. To'nought, if dh’ = dh; and the two al- 
titudes being measured by the same instrament, we shall 
have dh’ — dh = 0, when the errors will be those of the line 
of collimation, which, will happen in most cases. 
It will be reduced at least to very little. Suppose dh’ — 
Bite LO gah 19's vue on sin A = 57” sin A'= 98”, 
suppose A = 30° by a mean between the extreme values of 
sine A. 
If h alone has been observed, because A = A’, we shall 
then have ?’ = h+D; thus A’ is, supposed to be equal to 
h+D and dh’ to dh, therefore dh’ — dh = 03 in which 
case likewise 1 tang 2 = 0, there only remains dh cos A, 
which will never exceed dh. 
The errors dA become in the same manner: 
—— ,it will be reduced to nought, 
—dheos(A — 2) + dk’cosA —dhcos Acosx—dh sinAsinag+dh' cos A 
ee Sat ee 2S SS See ee = eae 
cos 6 sing sin aces @ 
sie (dk’ — dh) eos A + 2dh co8 A sin? hae dhs ain A ned (ut —dh) cos A 
= OO ——————— | = —- _—_ 
sim 2 Cos @ cos Q sin @ cos @ 
dh cos A tang sax dh sin ‘A 
cos @ cos 5 
At first glance, the error on ¢ and A,should increase in 
proportion as the sine x Jessens, and should be like infinite 
Te Ae SO a 2 the errors are reduced to 
dh cos A and oul A 
It might, therefore, be thonght that such is the error 
when vee to a maximum ; but it appears more natural 
to say that, whenever a 1s very small, the differential for- 
mula cannot give the exact effects produced by dh and dh’, 
I have to give an example of all these calculations. 
Let_us suppose, with Mr. Gauss, that ¢ = 8° 99’ 43731; 
0’ = 98° 2' 13”.4; § = 62° 38’ 26",9; h=h’ =45° 44’ 50”, 6. 
Calculation of the subsidiary Angle V dccordins lo Mr. Gaus’, 
Tang o..9°7263516 tang 6 0.4.6... + 45s 02845877 
' ORE iv anioiaiors do vee ce O'SL5B3I8 
C cos §. .0°33643293 C sin F—0......... 0°1852598 
0°0627839 tang V=62°36'58”,79 0°2856793 
tang F =49° 7’ 37,70 | ¢ 
§ 8 22 43,10 
F-¢= 40 44 54,60. . Trigo- 
