deviation itself is here so small. Excluding these four sets of observations, the mean value of so, deduced from the others will be 6° 47', and the difference between this and the values of so is not greater, in any case, than may be fairly attributed to small errors in the observations. It was not, however, my intention that the correctness of the theory should rest on the result of a single set of these experiments; and, although the computations were necessarily laborious, I had determined the co-ordinates for other values of p and q, so as to be able to undertake two or more sets of experiments of the same kind as the preceding: but having found considerable difficulty in adjusting the compass on the table, so that its centre should be at the proper distance from the centre of the table, and its north and south line be at the same time parallel to the meridian, and considering this mode, for that reason, liable to error, I resolved to compute afresh for the positions of the compass, according to a different adjustment.

The method which I now proposed to follow, was, to compute the distance of the centre of the compass from the centre of the table, and the angle made by this distance with the axis r or with the meridian line on the table, so that the ball should have a given position in the circle Ba. This may be deduced as follows from the equations (3), (6) and (7). Let r be the projection on the plane ry of the line p, joining the centres of the compass and ball; and the angle which this line makes with the axis x, which may be called the azimuth of the ball, p: or, which is the same thing, r the distance of the centre of the needle from the centre of the table, and the angle which this line makes with the meridian. Also let λ be the latitude of the ball, or the complement of the angle which the line joining the centres of the ball and needle makes with the line sCn: then y=r sin &; q=p sin x.

x = r cos p,
p = p cos λ,

Having obtained from this equation, then the equation (8) gives

Substituting the values of p and q in the equation (7),

≈ = p. (cos λ sin l cos d + sin λ sin d).

(12.)

From the equations (10), (11), (12), I computed the values of 4, r and z for the values of l 80°, 70°, 60°, &c. round the circumference; first, when was equal to 30°; secondly, when was equal to 45°; and lastly, when λ was equal to 60°. In all these I assumed p = 18 inches, wishing, for the reason I have beforementioned, to have the compass farther removed from the ball than in the foregoing experiments.

I now made the observations in the following manner: every 10° of the circumference of the table being divided into four equal parts, and lines drawn from the centre to the points of division, I divided the intermediate arc, with a radius of 20 inches, into quarters of a degree on a scale; and by this means, any angle could easily be set off very correctly to within about 5'. In this manner I obtained any particular value of 4, on one side of the meridian line reckoning from the south, and on the other side from the north; and by stretching a fine line across the two points, by

means of weights hanging over the edges of the table, I had the true direction of the centre of the ball, for that observation. The compass was placed on this line, so that its north and south line exactly coincided with it, and its centre was at a distance from the centre of the table equal to the value of r, corresponding to this value of : the ball was then lowered so that its height above the needle was the corresponding value of z. I now observed the angle which the needle made with the north and south line of the compass, and, as before, made my observations at both ends of the needle, and likewise on the contrary side of the meridian, taking the mean of the four observations, as the angle indicated by the needle: the difference between this and the angular distance of the compass from the meridian, that is, the value of 4, gave me the mean deviation caused by the action of the ball in that particular position. The value of so was computed from this deviation as before. I now also took precautions that the centre of the ball should descend, as nearly as possible, in the vertical passing through the centre of the table.

The following tables exhibit the results of these three sets of experiments.

I.

Latitude of the ball, λ = 30°, distance p = 18 inches, diameter of the ball = = 12.78 inches.

170

60

50

40

30

20

value of

Φ.

2°.20'

2

Computed value of so.

3°. 42′

3.48

3.32

80° N 13°.17 11.781 13.608 16°.00 15°.40 15°.30 15°. 20' 26.16 12.047 13.374 31.20 30.50 31.00 30.50 4.44 38.43 12.461 12.990 45.50 45.30 45.30 45.30 6.52 3.51 50.31 12.983 12.470 58.55 58.25 59.00 58.50 8.16 3.43 61.39 13.569 11.829 71.05 70.40 71.15 71.00 9.21 3.38 72.10 14.181 11.086 82.15 82.00 82.20 82.00 9.583.33 82.9 14.787 10.264 92.40 92.20 92.40 92.30 10.2343.30 10 91.42 15.358 9.387 102.25 102. 5102.30 102. 2010.38 8.484 111.20111.00 111.20111.10 10.17 7.580119.40 119.30 120.10 120.00 9.56 6.704 128.00 127.50 128.00 127.40 9. 8 3.28 5.882135.45 135.45 135.55 135.35 8.16 3.29 5.139 143.15 143.15 143.25 143.05 7. 3 3.25 144.54 17.426 4.498 150.50 151.00 151.10 150.50 6. 33.35 153.38 17.550 3.977 158.00 158.20 158.30 158.15 4.3843.34 70 162.24 17.633 3.594 165.30 165.40 165.40 165.25 3. 9 3.36 171.12 17.694 3.359172.45173.00 172.55 172.40 1.38 3.42

100.55 15.876 10° S 109.54 16.327

20

118.44 16.705

2

3.28 3.30

Mean 3.36

60

31.47
46.01

8.265

15.990 36.55 36.35

36.45

36.55

5.00 3.59

50

40

30

20

8.884 15.677 53.20 53.00 53.15 53.207.1244.00 58.52 9.558 15.253 67.55 67.45 67.40 67.40 8.63 3.57 70.27 10.347 14.729 80.45 80.30 80.20 80.20 10.014 3 52 14.122 92.15 92.05 91.40 91.35 10.544 3.51 13.451 102.20 102.10 102.15 102.00 11.29 3.52 12.736 111.50 111.30 111.30 111.25 11.453.54 11.998 120.00 120.00 120.00 120.00 11.32 32 3.54 11.260 128.00 127.45 128.10 127.50 11. 83.57 10.545 135.20 135.30 135.00 135.00 10. 16 3.56

80.59 11.161 90.42 11.961 10 99.48 12.720 0 108.28 13.419 10°S 116.48 14.043 20 124.56 14.589 30 132.55 15.051 40 140.49 15.432 50 148.40 15.733

818

156.30 15.960 70 164.20 16.121 80 172.10 16.216

1

9.874 142.15 142.20 142.00 142.15 9.17 3.58
9.267 148.40 149.00 148.40 148.30 7.58 3.56
8.743 155.15 155.35 155.20 155.00 6.374 3.58
8.319 161.20 161.30 161.04 161.30 5.00 3.54
8.006 167.35 167.55 167.40 167.30
7.814 173.45 174. 5 173.50 173.40

3.20 3.50 1-.40 3.47

Mean 3.54