632 GAUSS’S OBSERVATIONS OF THE 
axis of rotation with the plane of the circle relatively to the 
centre of the graduation expressed in parts of arc of the imner 
periphery of the circle, x being parallel with the diameter, pass- 
ing through the two zero points, and positive towards the right, 
y being parallel with the diameter, passing through the two 
points of 90°, and positive towards the upper part ; further, 180° 
—z being the angle between the two planes passing through the 
axis of rotation and the points A and B, (it being understood 
that the needle being supposed horizontal and its marked side 
being uppermost, the reckoning is from left to right from A to B,) 
and lastly, the mean between the two readings being designated 
by 7; the difference between them (understanding that the lower 
reading is subtracted from the upper reading) will be 
=2asnl+ 2ycosl +z, 
where the upper sign applies when the marked face is to the 
front and A is at the same time uppermost (or in this part of 
the world a South Pole), or when the unmarked face is to the 
front and B is uppermost; in the two other cases the lower 
sign applies. 
The above observations thus give sixteen equations, from 
which, by the method of least squares, 
ys SBS 
y = + 1532 
ore eens sty 
The comparison then gives (the arrangement being according 
to the amount of /)— 
1. Observation.| Calculation. Error. 
i il Ui 
67 28 45 | +102 | +122 | —20 
67 36 7' 4105 | £321 |- —16 
67 45 15 | — 48 — 30 —18 
68 2 42 — 18 ree +14 
89 51 24 + 24 0 + 24 
89 52 45 + 12 re + 13 
89 54 6 + 24 =i + 25 
89 58 10 = 45 aor — 44 
90 11 30 — 120 83 + 33 
90 12 10 — 153 —— ie 
90 18 34 — 147 — 153 + 6 
9015 7 | — 1385 — 153 + 18 
112 619] — 93 —111 +18 
112 17 42 —114 — 112 — 2 
1/2 24 57 | — 288 — 264 — 24 
112 36 22) — 297 — 264 — 33 
The sum of the squares of the remaining errors is 7924, whence 
