644 GAUSS’S OBSERVATIONS OF THE 
calculation according to the above formule, c and c’ must be 
computed with much more exactness than the absolute confi- 
dence due to them would in itself merit, otherwise the double 
determination for i, £, = would give but little accord*. How- 
ever other formulz not subject to this inconvenience but some- 
what less simple may be substituted ; these are subjoined, omit- 
ting the deduction, which is not a difficult one :— 
_ 2sin(f+e).sin (g—c) cos (Q+h) 
sin (f+ g).sim (h+c) * sin (Q— ce) 
_ _ 2sin (f!+¢).sin(g' —c) cos (Q + FH) 
~ sin (f’ + g/) . sin (2’ — ce’) * sin (Q — €) 
p= Sn (f* +49) - sin (f' + ec). sin (g' — c') . sin (A + c) 
~ sin (f’ + g').sin(f+c).sm(g—c).sm (A+ ec) 
II. The equation cos (2Q — B) = C (with the exception of 
the special case in which C = + 1), has always four different 
solutions or values of Q between 0° and 360°, in pairs differing 
180°. Such a pair of values of Q belong to one and the same 
tang 7 = 
value of i, but to opposite though equal values of £ > Jr now, 
as these latter quantities must from their nature be positive, one 
value of Q in each pair drops out of itself. But if one value of 
Q gives the signs of 4, ma opposite to each other, it is clear 
that the whole pair is to be rejected ; and if the same thing should 
take place with both pairs, it can only be concluded that the 
errors of observation render the combination of equations (1—6.) 
quite unfit for determining the unknown quantities. In our ex- 
ample the calculation gives the two following systems of values :— 
First System. 
te) 4 “l 
12 44 41 
Q 455 44 41 
i= 67 41 33 
4 — F 00051395 
m 
fe 
7 = ¥ 0:0042073 
Second System. 
a= 179 57 42 
— 1359 57 42 
jt) GO. eae 
4 = F 03443905 
ne 
4 = + 0°3563855 
~* All the calculations in this memoir have been made with the greatest pre- 
cision ; but fractions of seconds have been omitted in printing. If, therefore, 
any person should pursue the calculation with the intermediate values thus 
curtailed, he may sometimes find results which deviate a little. 
