642 GAUSS’S OBSERVATIONS OF THE 
sin? cos (A + c) = — = cos (A Ine? ee ey 
sin (f! + ¢ —i) = — cos (f' + Q@) 4 Sees 
; Ma ay q 
sin (g! — cl — i)= — =f co8 Co + GY 9) 5 eter ied 
sin icos (W + c) = 1 cos (! + Q) ocean ice ila 
16. 
Theoretically considered these six equations are sufficient for 
determining the six unknown quantities ¢, c’, +, Ss Q, 7; and 
the solution of this problem may be allowed a place here, al- 
though it has no practical value, because the enormous influence 
of the unavoidable errors of observation on the final result ren- 
ders this proceeding quite unsuitable. 
By multiplying the equations 1, 2, 3 respectively by 
sin (g + h), sin(f — h), sin (f + g), and adding, we obtain, 
after some easy reductions, 
sin (f+ c).sin (g + hk) = sin (g — c).sin (hk —f), 
whence ¢ can be easily determined, and most conveniently, by 
means of the formula, 
tan (¢ + 3f—29) = — tan $ (f+)? . cot(h-—3f+ 3 9)- 
In a similar manner we obtain from the equations (4.), (5.), (6.), 
tan (¢ + 4/'— 3!) = — tang (/" +g). cot (! — 3/1 + 29). 
The numbers in our example are— 
f =67 26 11 fl = 67 56 11 
g = 67 43 46 g! = 67 35 35 
h= 89 51 49 h' = 90 10 48 
whence, according to the above formula, 
6 = +. Kegin c= — 14’ 18", 
The magnitude of these values appears already almost beyond 
probability, and the little confidence due to them becomes visible 
on developing the influence upon them of small errors in the 
fundamental numbers. The differential formula serving thereto 
may be given in several forms, of which the following is one :— 
_ _ sin(g — cc). sin (A + c) 
*°= ~ sin Wf). sin + 9) 7 * 
