650 GAUSS’S OBSERVATIONS OF THE 
i=f+iec—tecsf+usnf 
i=g—c—tcosg—using 
i=f'+c+atcosf’ —ausin/' 
ti=gi—c +atcosg + ausing’. 
The five unknown quantities i, c, c', t, uw cannot, it is true, be 
determined by four equations, but they may be expressed by one 
quantity which remains indeterminate ; and if c! — ¢ be selected 
for the purpose, we learn, in the clearest manner, in what degree 
we are justified in neglecting the quantity so represented. The 
elimination itself is in each separate case most conveniently con- 
ducted, after substituting the numerical values in the observa- 
tion data. 
In our example the four equations become 
i = 67° 26' 11"4 ¢ — 0:3837 ¢ + 0°9234 u 
i= 67 43 46 —c — 03790 ¢ — 09254 u 
i= 67 58 11 + e¢ + 03801 ¢ — 0°9393 u 
i= 67 35 35 — ce + 0:3862 ¢ — 0°9368 u, 
whence we find by elimination, 
67° 41! 54" — 0:0006 (c! — c) 
— 934 + 0°:0002 (c' — c) 
2 + 648 + 05369 (¢ —c) 
1(¢+c)= — 73 + 0:0037 (c' —c). 
Hence we see that the arbitrary supposition of the equality 
of c and c’ does indeed make a secure determination of u im- 
practicable, but that it has no sensible influence on the values 
of i and #, and only a small influence even on the determination 
of the mean value of ¢ and c’. 
The mean of these four equations is 
i = 67° 40’ 56” + 00009 ¢ — 0:0011 x, 
where the absolute part is the simple mean of f, g, f', g', and 
might fairly, without anything further, have been taken as the in- 
clination. This is in fact the ordinary mode of proceeding, and 
may be adopted without scruple where the values of f, 9, f’, g 
do not present any great differences. 
22. 
Before leaving the example which has been under treatment 
hitherto, I will remark further, that the equations (3.) and (6.) are 
susceptible of an abbreviation quite similar to that of the others. 
We may put 
il 
[-~ Le Ss 
