654 GAUSS’S OBSERVATIONS OF THE 
This form has the advantage, that the coefficients of x and y 
always differ little: from unity, and in fact with so little excen- 
tricity of the centre of gravity as in the four needles in question, 
and with such moderate fluctuations of n, we are justified in 
taking unity in lieu of those coefficients, which I term the ab- 
breviated calculation. I have however been at the pains of cal- 
culating the 192 coefficients more exactly, leaving out only the 
factor = although the principal use of doing so is only to 
show more clearly the admissibility of the abbreviated calcula- 
tion. In the sequel the non-accented letters N, 2, y will refer 
to needle I, and the values for the remaining three needles will 
be distinguished respectively by one, two and three accents. 
The values selected for the present calculation are,— 
lL =praao O- 
N = 5"-847785 
N’ = 5°686867 
N" = 6°181742 
IN 5*949567 
I do not subjoin the calculations themselves in full, on account 
of the space they would occupy, but only so much of them as is 
necessary to give a general view of the proceedings. The values 
of the coefficients which differ most widely from unity, are 
0°96895 and 1°04324 on the 9th and 16th of June with needle 4. 
25. 
From the two equations afforded by the observations of one 
needle on each day, two other equations, which we will call I. and 
II., may be formed, by halving both their sum and their differ- 
ence. There arise thus 48 equations I. and as many ILI., the first 
of each of which are subjoined as examples. The original equa- 
tions from the observations of the 20th May with needle 1 are,— 
i= 67 11 0 +c — 1:02880 2 + 1°00460 y 
i= 67 58 46 — c — 0'99473 x — 1:01038 y, 
whence there arise the deduced equations— 
i = 67° 34! 53" — 1:01176 # — 0°00289y. . (I.) 
c= + 1433! + 0:01703 a — 1:00749y ... (IL) 
In order to be able to institute the test pointed out in Arti- 
cle 8, I have added another member to the equations I. by writing 
