68 
ARCHDEACON PRATT ON THE ATTRACTION OF THE 
26. That the formula (8.) which I have been using- to determine these values does 
not, thus far, lead to material error may be shown by substituting the 42nd pair of 
values in the test given by formula (7-)- this case 
a=38°-86, (p=4°-314, ^ ^ = 20°'50=20° 30', 
and formula (7.) becomes 
^=log-‘ 
- 11-0379639 - 
9-5443253 
< 20-5822892 > 
19-9431752 
. 0-6391140. 
= 4°-356. 
Hence the error=4°-356— 4®-314=:0°-042, and this equals TSsi’d of the whole. 
If the same test be applied to the 40th values of a and <p, the error is only s^th of 
the whole ; and as we pass further back it becomes absolutely evanescent. 
27. The remaining values of a and <p, after the 42nd as above determined, we must 
find by solving equation (3.) by trial and testing the values by formula (7.). 
Horizontal attraction of the prism on the station of the observer 
4 D 180 
- ^1 + tan ^ 9 tan i sinw + ^1 +tani 9 tanl sin w' cos ^ 9. 
In this tan 1 9 tan - w and tan - 9 tan i uj' may he neglected as quite insensible ; for tan i ui and tan - w' can 
neither of them be greater than 1, and in that case 9=0; and when they have any other sensible value, 
tan i 9 is of insensible magnitude. So also as 9 is never made larger than 1°, or so large, in the application of 
this formula, cos i 9 may he put = 1 ; and the formula becomes 
Horizontal attraction of the prism on the station of the observer 
3 p /3 / / . • 
=- i !— — (sin w + sin cy'). 
4 D 180 ^ 
The values of I need not follow any law, but may be chosen in each lune according to the form of the verti- 
cal section ; some values being long and some short, according as the variations of w and w' are slow or rapid. 
The angles w and w' must be found as follows : — The lune having been divided into compartments, the average 
height and depth of the top and bottom of the prism standing on each compartment, above and below the 
observer’s horizon, must be divided by the horizontal distance of the prism from the observer. This will give 
the tangents of the angles cw and w', whence the sines may be found. 
The above expression must be thus calculated for all the compartments : the whole added together gives the 
attraction of the mass standing on the portion of the lune to which this method is to be applied. This sum, 
multiplied by the cosine and sine of the azimuth, will give the attraction in the meridian and in the prime ver- 
tical. The same being done all round the circle, the resultant attraction and the azimuth of the plane are easily 
found ; whence the deflection is known, and the various angles of observation may be corrected. 
In using the method of the text for the parts beyond, the heights must be measured from the sea- 
level, and not from the surface, parallel to the sea-level, passing through the station of the observer. This is 
done in the text in the case of Kaliana, because it appears that below the level of that place there are no varia- 
tions of surface sufficient to produce any sensible alteration in the attraction of the whole mass. This, how- 
ever, will not be the case with stations in the mountains. 
